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UNITED STATES OF AMERICA. 



MANUAL 



OF 



Geometrical and Infinitesimal 



ANALYSIS. 



v;- 



By b. sestini, S.J. 

AUTHOR OF ANALYTICAL GEOMETRY, ELEMENTARY GEOMETRY, AND A TREATISE ON ALGEBRA ; 
PROFESSOR OF MATHEMATICS IN WOODSTOCK COLLEGE. 



/ 




BALTIMOKE: 

PUBLISHED BY JOHN MURPHY & CO. 

\ 

No. 182 Baltimore Street. 

1871. 






Entered, according to Act of Congress, in the year 1871, by 

JOHN MURPHY & CO., 

in the Office of the Librarian of Congress, at "Washington. 



PREFACE. 

nnHIS manual, prepared with the view of its serving as an 
introduction to the study of Physical Science, was only 
intended for a class of students intrusted to the care of the 
compiler. The suggestion of friends that the work might 
prove advantageous to others induces him to offer it to the 
public. 

Works of analysis — some of them voluminous — are not 
wanting ; nor does our little book pretend to give a complete 
development of its subject. For this reason we call it a 
manual, which excludes all discussions the results of which 
are seldom or never called into use in the applications. It 
is hoped, however, that it will sufficiently serve the purpose 
intended. 

A detailed Index will contribute to render the manual more 
useful. It will also give a better idea of the nature of this 
little work. 

We leave it to the reader to judge whether, without detri- 
ment to lucidity, our efforts to combine comprehensiveness 
with brevity and exactness have been successful. 

B. Sestini, S. J. 

Woodstock College, Md., January 18, 1871. 

iii 



HHgr^ N. B. When Theorems of Algebra, Trigono- 
metry y &c., are mentioned, reference is made to books 
previously published by the author of this Manual. 



IV 



CONTENTS. 



PRINCIPLES OF ANALYTICAL GEOMETRY. 



PAGE 

I. Rectilinear and polar co-ordinates of a point on a 

plane surface 9 

II. Orthogonal and oblique co-ordinates 10 

Formulas (1) to pass from one to another system of axes 10 

Formulas (2) to pass from the orthogonal to the polar co- 
ordinates 10 

III. Equation of the straight line 10 

General equation (3) 11 

Equation of the line passing through the origin of the axes... 11 

Equation of the line passing through a given point 11 

Equation of a parallel line 12 

Equation of the perpendicular 12 

Criterion of perpendiculars 12 

IV. Equation of the circle — referred to the centre 12 

" " referred to the extremity of a 

diameter 13 

" " referred to any origin 13 

V. Equation of the parabola — definitions 13 

" " referred to the vertex 14 

VI. Analysis of equations — geometrical loci 14 

VII. Properties of the parabola 15 

The parameter is third proportional after the abscissa and the 

ordinate of any point , 16 

The parameter is equal to the double ordinate passing through 

the focus 1G 

VIII. Tangent and other properties of the parabola 16 

The parallel to the axis and the radius-vector form equal 

angles with the tangent 17 

The normal bisects the angles formed by the same lines 17 

IX. Equation of the parabola referred to different axes 17 

X. Polar equation of the parabola 19 

v 



VI CONTENTS. 

PAGE 

XI. Equation of the ellipse — definitions 19 

" " referred to its axes (1) 20 

Second form of the same equation (8) 21 

XII. Analysis and corresponding geometrical locus 

of the last equation 21 

Characteristic property of the ellipse — the sum of the ra- 
dius-vectors is equal to the transverse axis 22 

XIII. Parameter of the ellipse 23 

The parameter is third proportional after the transverse 

and the conjugate axis.. 23 

XIV. Tangent and normal 23 

The normal bisects the angle formed by the radius-vectors 23 

The radius-vectors form equal angles with the tangent 23 

XV. Diameters of the ellipse — Conjugate diameters 23 

Condition to be verified with regard to conjugate diam- 
eters (3) 25 

Length of conjugate diameters (4).... 26 

XVI. Equation of the ellipse referred to conjugate 

diameters 26 

XVII. Polar equation of the ellipse 27 

XVIII. Theorems — 1st. The sum of the squares of the axes is equal 

to the sum of the square of the conjugate diameters 28 

2d. The parallelogram on the conjugate diameters is equiv- 
alent to the rectangle on the axes 29 

XIX. Equation of the hyperbola referred to its axes... 29 

Its different forms (1), (3) 30 

XX. Analysis and corresponding geometrical locus 

of the equation 31 

XXI. Parameter of the hyperbola 31 

The parameter is third proportional to the transverse and 

the conjugate axis 32 

XXII. Tangent and normal 33 

The angles formed by radius-vectors are bisected by the 

tangent and by the normal 33 

XXIII. Asymptotes of the hyperbola 33 

XXIV. Equation of the hyperbola referred to the 

asymptotes 34 

XXV. Polar equation of the hyperbola 36 

XXVI. General polar equation 36 



CONTENTS. Vll 

PAGE 

XXVII. Equation of the cycloid 37 

XXVIII. Rectilinear and polar co-ordinates of 2>oints in 

space 38 

XXIX. Equation of the plane 40 

XXX. Equations of the straight line in space 41 



PRINCIPLES OF INFINITESIMAL CALCULUS. 

PAET I. 

DIFFERENTIAL CALCULUS. 

I. Infinitesimal quantities; different orders and 

expressions of the same 43 

II. Functions 44 

III. Differentials . 45 

First and second state of a function 46 

IV. Preliminary theorems 46 

1st. The binomial (1 -f- w ) irx which o» is infinitesimal raised 

to the infinite power— gives for result e = 2, 7182818 ... 48 

CO 

2d. The binomial (1 — cj) raised to the negative infinite 

power — gives for result e = 2, 71 48 

3d. The ratio between an infinitesimal arc and its sine is 

equal to 1 49 

V. Differentials of algebraic functions 49 

i. Differential of a db z 49 

11. " " az 49 

in. " " — 50 

z 

iv. " " z° 50,51 

VI. Differentials of transcendental functions 52 

1. Differential of Iz 52 

11. « <■<■ a z 53 

Hi. " " sin z 53 

iv. " " cos z 53 

V. " " tg z 54 

vi. " " cotz 54 

vii. " tl arc (sin = z) r -" 



VU1 CONTENTS. 

PAGE 

Tin. Differential of arc (cos = z) ■. 55 

ix. " " arc (tg = z) 55 

x. " " arc (cot =. z) ,.. 55 

VII. Differentials of the sum, product, and quotient of 

different functions of the same variable x 55 

i. Differential of the sum 56 

ii. " " product 56 

in. " "quotient 57 

VIII. Successive differentials and their orders . 57 

i. Successive differentials of y = x n 58 

ii. I" " " y = a x 58 

in. " " " y = sin x 59 

iv. " " "y — \og(x) 59 

IX. Derivative functions and their orders 59 

The derivative function of any order is given Toy the ratio 
between the corresponding differential of the primitive 
function and the corresponding power of the independent 

variable 60 

The differential of any order of a given function is equal to 
the product of the derivative of the same order by the cor- 
responding power of the differential of the independent 

variable 60 

X. Maclaurin 9 s formula 60 

Its applications, i. to f (x) ■= e 62 

ii. to / (x) = sin x 62 

in. tof{x) = cos x 62 

XI. Taylor's formula 62 

Applied, 1st, to f (x) = s/ x 63 

2dly, to/(x) —-L 64 

vx 

XII. Maxima and minima of functions of a single 

variable 65 

Conditions to be verified in order that a function may be a 

maximum or a minimum 65 

Taylor's formula reveals whether and when these conditions 

are verified 65 

XIII. Values of functions which assume an undeter- 
mined form 67 

1st, of functions which assume the form ^ 68 

2d, " " " " ~ G9 

3d, " " " " O-oo 70 

4th, " " " the forms 0°, oo°, 1" .70. 71 



CONTENTS. IX 

PAGE 

XIV. Chord of an infinitesimal arc of a continual curve 72 

XV. Tangent , subtangent, normal, and subnormal of 

any plane curve 73 

Applications — 1st, Functions of the parabola 75 

2d, Functions of the ellipse 76 

3d, Functions of the hyperbola 77 

XVI. Differential total and partial of a function of 

different independent variables 77 

The total differential of a function of different variables is 

equal to the sum of the partial differentials 79, 80 

XVII. Derivative functions 80 

Different manners of representing them 80, 81 

They are the same whatever be the order of differentiation 81 
Formulas expressing the successive differentials of a func- 
tion, of different variables 82, 83 

XVIII. Singular point of plane curves 83 

Example of a double 'point 85 

Example of a cusp or point of regress 86 

Example of an isolated point 86 

XIX. Convexity and concavity ; points of inflection 86 

Criterion to determine when the curve turns its convexity 

or its concavity to a given axis 87 

Criterion of the points of inflection 88 

Example 89 

PART II. 

INTEGKAL CALCULUS. 

XX. Indefinite integrals 90 

Complete and incomplete integrals 90 

XXI. General Theorems 91 

1st. Constant factors of a given differential may be pla^«a 

outside the sign of integration 91 

2d. The integral of the sum of various differ, . 'ials is equal 

to the sum of the integrals of each term » 91 

3d. The integral of z m dz is obtained by suppressing dx, aa^_ 

ing 1 to the exponent, and dividing x m + 1 by m -f- 1 91 

XXII. Immediate integration ; integration by substitu- 
tion; integration by parts 92 

i. Immediate integration; examples 92 

ii. Integration by substitution ; examples 92 

in. " by parts ; " 94 



X CONTENTS. 

PAGE 

XXIII. Definite -or limited integrals 97 

The definite integral of F (x) dx taken between two deter- 
mined limits is the sum of all the values which F (x) dx 
receives by the infinitesimal variations of x from one 
limit to the other 98 

The integral of the same differential expression is given 
by the difference of the values which its indefinite in- 
tegral, abstraction being made from the constant, re- 
ceives when we substitute in it the values of x corre- 
sponding to the two limits 99 

Examples 100 

XXIV. Differentials of an arc, of an area terminated 

by an arc, of a, sector, and their correspond- 
ing integrals 101 

i. Differential of the arc; its integral; examples 102 

ii. Differential of an area; its integral, and examples 104 

in. Differential of a sector; its integral, and example 106 

XXV. Circular curvature ; osculatory circle and ra- 
dius of curvature of aplane curve 107 

XXVI. Evolutes and involutes 109 

Examples — i. Evolute of the parabola Ill 

ii. Evolute of the cycloid 113 

XXVII. Integration by series 114 

By this method of integration some functions can be devel- 
oped into series — 1st, Log (1 -f- x) 115 

2d, Arc (sin = x) 116 

3d, Arc(tg = z) 117 

XXVIII. Integration of differential equations of the first 

order and degree, bettveen two variables 117 

Inexact differential 120 

Resolution by multiplication. 1st case 120 

2d case 123 

Resolution by separation 124 

1st case 125 

2d case 126 

XXIX. Integration of linear differential equations of 

the first order, containing only two variables 127 
XXX. Integration of linear differential equations of 

the second order, and bettveen tivo variables 128 



PEINCIPLES OF 

ANALYTICAL GEOMETRY. 



I. Rectilinear and polar co-ordinates of a point on a plane 

surface. 

Let XX', YY' (Fig. 1) be two indefinite straight lines 
cutting each other in A and forming any angle. Take any 
point M on the plane of the two lines, or, as they are called, 
axes, and from M draw MH, MK parallel to the axes. The 
position of the point M is evidently determined relatively to 
the axes by these parallels, which are called rectilinear co-ordi- 
nates of the point M. MH, or its equal AK, represented by 
x, is called the abscissa, and MK, represented by y, is called 
the ordinate. XX' is the axis of abscissas, YY' the axis of 
ordinates, A the origin of co-ordinates. The axes are called or- 
thogonal when they cut each other at right angles ; otherwise, 
oblique. In every case, taking for positive the abscissas from 
A toward X and the ordinates from A toward Y, the abscissas 
from A toward X r and the ordinates from A toward Y r must 
be considered negative. It is thus plain that, varying x and y 
from — oo to + oo, we may, by means of them, determine 
the position of all and each of the points of the plane. 

Polar co-ordinates offer the same advantage. Let A be a 
point on the plane, and let XAX' be a fixed straight line or 
axis on the same plane and passing through A. Let again M 
be any point of the plane, and AM, represented by p, be 
its distance from A. The angle MAX, which we call w, 
together with p, determines the position of the point M on the 
plane ; and these are called polar co-ordinates of the point M. 



10 PRINCIPLES OF ANALYTICAL GEOMETRY. 

The point or centre A is called pole, the fixed line XAX' 
polar axis, the distance AM radius vector. To determine the 
position of all and each of the points of the plane, it is neces- 
sary to vary the radius vector from to go and the angle w 
from to 360°. 

II. Orthogonal and oblique rectilinear co-ordinates. 

Let (Fig. 2) AX, AY be orthogonal axes, and A'X', A'Y' 
oblique axes. Any point M referred to the first system shall 
have x = AK and y = KM for its co-ordinates, and, referred to 
the second system, x' = A'K r , y' = K7M. Call x , y the co- 
ordinates AB, BA r of the origin A! referred to the orthog- 
onal axes. The angles which A'X', A'Y' form with AX are, 
for brevity sake, represented by (x'x), (y'x). Draw now from 
A' and K', A'C, K'D parallel to AX, and K'C parallel to AY, 
we shall have 

x = AB + A'C + K'D, y = BA' + CK' + DM ; 
x = x -f x ! cos (x'x) + y' cos (y'x) 
y = y -f- x' sin {x'x) + y f sin (y'x). 

By means of these formulas, which give us the orthogonal 
co-ordinates by the oblique, we may pass from one to another 
system of axes. 

Orthogonal and polar co-ordinates. 

Let (Fig. 2) A be the pole, and AX the polar axis, and M 
any point on the plane XAY, having AM = p for radius vec- 
tor, and MAK = w for corresponding angular co-ordinate. 
From the triangle MKA, right-angled in K, we obtain 

f X = p COS 6J 

2 • 

I y = p sin w. 

And by means of these formulas we may pass from the or- 
thogonal to the polar co-ordinates. 

III. Equation of the straight line. 
As each point on the plane of the rectilinear axes has its 



i. e., (1) { 
I: 



PRINCIPLES OF ANALYTICAL GEOMETRY. 11 

co-ordinates, so the different points of a line described on that 
plane have their own co-ordinates. Now it happens that the 
value of the ordinate, given by the corresponding abscissa, is 
found to be given under the same form for each point of the 
line; for instance, we may find y = ux, or, more gener- 
ally, y — f[x) for each and all points of the line referred to 
the axes ; y = f(x) is, in this case, called the equation of the 
line or of the geometrical locus described on the plane of the 
axes, whatever the form of the line may be. To give an ex- 
ample : Let (Fig. 3) RE, be a straight line on the plane XOY 
of the orthogonal axes, and let x = OK, y = KM be the co- 
ordinates of one of its points M. We shall have = 

AK 

&7MAX, and calling a this tangent, 

■ — . ^ = a or y = ax -f- a • AO. 
x + AO 9 

Now we have from the right-angled triangle ABO, BO = 
a ' AO. BO is the ordinate of RR corresponding to the origin 
of the axes, and which we represent by b. Hence 

(3) y = ax + b. 

But M is any point of RR' ; hence this equation of the first 
degree represents the corresponding co-ordinates of any point 
of the straight line, and is, consequently, the equation of' the 
same line. We infer from this, 

1st. If the straight line passes through the origin of the 
axes, the equation of the line is 

y — ax. 

2d. If the same line, whether it passes through the origin 
or not, passes through a point, C, for instance, having x, and 
y, for its co-ordinates, together with the above equation, we 
shall have y f = ax, -f- 6 or y, = ax, ; and in both cases 

y — y, = a(x — x,) 
the equation of a straight line passing through a given point 

fa> y,). 



12 PRINCIPLES OF ANALYTICAL GEOMETRY. 

3d. If B/B/ represents another straight line parallel to RR, 
calling b f the ordinate OB', corresponding to the origin, since 
B'A'X = BAX, the equation of the parallel will be 

y = ax -f- b' . 
4th. If NN' is drawn perpendicular to BR, calling b' f the 
ordinate, OI corresponding to the origin, its equation will be 
y = tgWtX'x + 6". 

But on account of the right-angled triangle ACL, and be- 
cause ^N'LX = — tyN'LO, 

foN'LX = — cofMAO = J-r^ = — - ; hence 

1 

y = x 4- b" 

is the equation of a straight line perpendicular to one repre- 
sented by y — ax -f b. It follows, therefore, that in order 
that two straight lines represented by the equations 

y = mx 4- c, y = m'x + c f 

be perpendicular to one another, it is necessary and sufficient 

that m'= , or that the equation 

■ m 

mm! + 1 = 

v 

be verified. 

IV. Equation of the circle. 

Let r be the radius of the circle AMB (Fig. 4), C the cen- 
tre, which is at once the origin of the orthogonal axes CX, 
CY. Let M be any point of the periphery, and CK, KM 
the co-ordinates x, y of that point. Drawing the radius CM, 
the right-angled triangle CMK gives us 

y 2 = r 2 — x 2 , 
which being verified for any point of the periphery, is the 
equation of the periphery, referred to the orthogonal axes hav- 
ing their origin in the centre. 

But let the origin be the extremity A of the diameter AB, 



PRINCIPLES OF ANALYTICAL GEOMETRY. 13 

the axes AX, AY being still orthogonal, and AK, KM the 
co-ordinates x and y of the point M. Now MK is mean geo- 
metrical proportional between AK, KB, and therefore y 2 = 

x (2r — x), or 

y 2 = 2rx — x 2 ; 

which being verified for any point of the periphery, is conse- 
quently the equation of the periphery, referred to the orthogo- 
nal axes having their origin at the extremity of a diameter. 

But let the origin of the orthogonal axes be anywhere, for 
instance in O, and let OH, HM be the co-ordinates x, y of 
any point M of the periphery referred to the orthogonal axes 
OX, OY, and parallel to CX, CY. Call a and b the co-ordi- 
nates OD, DC of the centre of the circle. We have, from the 



right-angled triangle CKM, CK + MK = CM ; but CK = 

OH — OD = x — a, MK = MH — DC = y — b and 

CM = r ; hence 

(» - af + (y - bf = r\ 

an equation of the second degree, like the preceding, and which, 
being verified for every point of the periphery relatively to the 
axes OX, OY, is therefore the equation of the periphery, re- 
ferred to these axes. 

V. Equation of the parabola. 

Let (Fig. 5) the straight lines DD', CX be perpendicular to 
each other, take a point F anywhere on CX, and let A be the 
middle point of CF. Let now the curved line MHAM' pass 
through A, and as the point A is equally distant from F and 
DD r (the distance of a point from a straight line is known to 
be the perpendicular from the point to the line), so let all the 
points of MHAM' be equally distant from DD' and from F. 
This curve line is the parabola, the point F is called the 
focus, DD' is called directrix, CX the axis of the curve, and 
the point A of intersection between the axis and the curve 
the vertex. 

Let now A be taken for the origin of the orthogonal axes 



14 PRINCIPLES OF ANALYTICAL GEOMETRY. 

AX, YAY', to which we refer the curve, and let AK, KM be 
the co-ordinates of any of its points M. Join F with M, and 
let the variable FM, called radius vector, be represented 
by p, and let the constant CF be represented by p. We shall 
have first, 

AF = AC = ip; 

and since MF is equal to MN perpendicular to DD', and 
MN = AK + AC, 

P = x + ip, 
from which p 2 = x 2 + px -f- Jp 2 , 



but from the right-angled triangle MKF MF = MK + 
(KA — AF) 2 ; hence also 

p 2 = y 2 + % 2 — p x + \p 2 . 

From this and the preceding value of p 2 we obtain easily 

y 2 = 2px, 

which, being verified for each and all the points of the curve, is 
the equation of the parabola referred to the orthogonal axes 
having their origin in the vertex, the axis of the curve being 
axis of abscissas. The constant 2p is called parameter or 
measure of the curve. In fact, it is plain from the above 
equation that, supposing the same values for x, the branches 
of the curve will open more or less according to the magni- 
tude of the parameter. And indeed, from the analysis of the 
above equation we may infer the properties of the curve. 

VI. Analysis of equations and geometrical loci. 

To find out the curve or the geometrical locus to which a 
given equation belongs, and to find out from the same equa- 
tion the properties of the corresponding geometrical locus, is 
called the analysis of the equation, and the name of analytical 
geometry is accordingly given to the branch of science which has 
for object this analysis. This process is evidently the inverse 
of the preceding, as shown in the foregoing examples. 



PRINCIPLES OF ANALYTICAL GEOMETRY. 15 

The equation y 2 = 2px of the parabola, already obtained, 
may also be written as follows : 

V = =*= 4/2^, 

in which p is positive. To find out the geometrical locus of 
this equation draw the axes XX', YY ; at right angles and 
intersecting each other in A. Take then from A different 
abscissas x, and substituting their values in the last equation, 
the resulting values for y will be the ordinates corresponding 
to the abscissas and marking the geometrical locus with their 
extremities. It is plain, 1st, that with x — 0, y also = 0. 
2d. No real ordinates correspond to negative abscissas. 3d. 
Two real ordinates correspond to each positive abscissa, equal 
in length, but opposite in sign, and these ordinates increase 
with x from to 00 ; i. e., the geometrical locus corresponding 

to the equation y — dz /2p' xisa curve which cuts the axis 

of abscissas at the origin and touches the axis of ordinates 
at the same point; it has a double branch, one on each side of 
the positive axis of abscissas, and equal to one another, depart- 
ing more and more from this axis as they do from the axis 
of ordinates by their increase. 

VII. Proper-ties of the parabola. 

It follows besides, that AX, called also axis of the curve, 
bisects all the chords parallel to the tangent of the vertex A. 
Taking from A on XX', AF = AC == Jp, and drawing 
from C, Diy the directrix parallel to YY r , it follows that 
each point of the curve is equally distant from the focus and 
the directrix. In fact, MF* = MK + (AK — AF) 2 = y 2 
+ (x — }p) 2 = 2px + x 2 + lp 2 — px — x 2 + px -f lp 2 = 
(x + ip) 2 ; hence MF = x + Jp. But MX perpendicular to 
DD r = KA + AC = x + ip] hence MF = MX. 

Resuming again the equation y 2 = 2px, observe that it is, 
resolvable into the proportion 

x:y::y:2p; 



16 PRINCIPLES OF ANALYTICAL GEOMETRY. 

i. e., the parameter in the parabola is a third proportional 
to any abscissa and the corresponding ordinate. 

Hence, calling q the ordinate drawn from the focus, since 
then \p : q : q : 2p, and consequently q = p, it follows that 
the parameter of the parabola is equal to the double ordinate 
passing through the focus. 

VIII. Tangent and other properties of the parabola. 

Join (Fig. 6) the focus F with N, the foot of the normal 
drawn to the directrix from any point M of the curve. Join 
also F with M, and draw from M, ME perpendicular to NF, 
produced to T, to meet the axis AX of the curve, as also on 
the opposite side toward P. This line TP is the tangent of the 
point M of the parabola. To show this, it is enough to prove, 
first, that none of the points of TP are equally distant from 
the focus and the directrix except M ; secondly, that all the 
points of TP on either side of M are outside of the branch. * 

Because MF = MN and ME is perpendicular to NF, 
the triangles MEN, MEF are equal • hence NE = EF, and 
drawing from any point P of TP,PN, PF, these two oblique 
lines also are equal to one another. But drawing from P, 
PQ perpendicular to the directrix, since PQ < PN,it follows 
also that PF > PQ. The point P is then not equally distant 
from the focus and the directrix, and thereby not on the curve. 
Now such a point may be within the branches of the parabola 
or outside of them. In either case drawing from it a perpen- 
dicular to the axis AX of the curve, this perpendicular has all 
its points equidistant from the directrix, but only one of them 
is at once equidistant from the directrix and the focus, and 
this one is the point of intersection with the curve. All the 
points between the curve and its axis are nearer to, and all 
those outside of the branch are farther off from the focus than 
the directrix. But PF > PQ ; hence all the points of TP 
on either side of M are outside of the branch, and TP touches 
the curve in M. 



PEINCIPLES OP ANALYTICAL GEOMETRY. 17 

Produce NM to X', the angle PMX' = MTX ; but PMX' 
= TMN = TMF, hence the angles at M and T of the trian- 
gle MTF are equal to each other, and consequently FM == 
FT. But FM = MN = CK j therefore TF = CK, and con- 
sequently TC = FK. Now AC = AF (= |p), hence AT 
= AK ; but AK is the abscissa x of the point M, hence TK 
= 2x. The segment TK from the point T of the axis met 
by the tangent, to the point of the same axis corresponding to 
the ordinate of M, the point of contact, is called the subtangent 
of that same point. Therefore the subtangent of any 'point of 
the parabola is equal to twice the abscissa of the same point. 

It follows from this that we may draw a tangent to any 
point M of the parabola by drawing first a perpendicular to 
the axis from that point, and taking on the axis a point twice 
the distance from the foot of the perpendicular than the vertex 
of the curve is. The straight line joining this point with M 
touches the curve on that point. 

We have remarked in the preceding process that the angles 
PMX', TMF are equal to each other ; i. e., 

For any point of the parabola the parallel to the axis and the 
radius vector form equal angles with the tangent 

Draw now from M, MR perpendicular to the tangent. The 
segment MR of this perpendicular, between the point of con- 
tact and the axis, is called the normal of that point. Now 
from the equality of the last-mentioned angles it follows that 
FMR and RMX' are also equal ; i. e., 

The angle formed by the radius vector of any point of the 
parabola and a parallel to the axis drawn' from that point, is 
bisected by the normal. 

IX. Equation of the parabola referred to different axes. 

Taking (Fig. 5) the origin of the axes, in the focus F, and 
considering FX as the negative axis of ordinates, and FX, 
perpendicular to the first as positive axis of abscissas, repre- 
senting besides by x„ y, the abscissas and ordinates of the new 



18 PRINCIPLES OF ANALYTICAL GEOMETRY. 

system corresponding to any point H of the curve, whose 
co-ordinates x y y relatively to the first system of axes are 
AB, BH ; since AB = AF — FB and BH is ordinate with 
regard to the first, and abscissa with regard to the second 
system, we shall have 

x = |p — y f and y = x,. 
These values substituted in the equation of the parabola y 2 = 
2 px referred to the first system of axes, will give us 

x f 2 = p 2 — 2 py r 

Let us now take a diameter for axis of abscissas and the cor- 
responding tangent for axis of ordinates. 

A parallel A'X ; (Fig. 7) to the axis AX, drawn from any 
point A' of the parabola, is called diameter. Take A'X' as 
axis of abscissas and TY' tangent in A' as axis of ordinates, 
and let the co-ordinates of the curve referred to the new sys- 
tem be represented by x', y'. Call x , y the co-ordinates 
AB, B A! of the origin A! of the new axes referred to the first 
system of axes and a the angle Y'TX, which the tangent Y'T 
forms with the axis of the parabola ; we shall have (II.) (x f x) 
= 0, (y'x) = a, and 

x = x + x f -f- y' cos a, y = y -f y 1 sin a. 

Now, from the right-angled triangle A'BT, A'B or y = 

A'T sin a, and BT — A'T cos a, hence ^ = tg a, and y Q 

= BT tg «, but (VIII.) BT = 2x ; hence 

y Q = 2x tg a. 

Besides (V) y 2 = 2p# , therefore, dividing this formula by the 

preceding, 

p p cos a 

tg a sin a ' 

2 

and since, from y\ = 2px 0> x<> = |^ ; we have also 

zp 

p cos 2 a 
2 sin 2 a ' 



PRINCIPLES OF ANALYTICAL GEOMETRY. 19 

p cos 2 a , . , , p COS a , , . 

hence , x = -/___ +y+y«^* -^— + y sm «. 

Substituting these values in the equation y 2 = %?a? of the para- 
bola referred to the original axes, we obtain 

./2 _ o _P „r 



y'* = 2 -^— x'. 

J Sin z a 



10 



From which, calling p f the constant factor^. 



sm a 

,/2 _ On'ff 



y n = 2p r x'. 

The coefficient 2p' is called parameter relatively to the diam- 
eter A'X'. It follows from this equation that all the chords 
parallel to the tangent TA'T are bisected by the diameter. 

X. Polar equation of the parabola. 

Take (Fig. 5) the pole in the focus, and, for polar axis, the 
axis of the curve from F toward the vertex A. Let M be 
any point of the curve. The polar co-ordinates of this point 
are p = FM and MFA = w. Now MF equal to MN, perpen- 
dicular to the directrix, is also equal to KA + AC = x + ip, 
hence p = x + \p> 

Now x = AF + FK = \p + p cos MFK = \p — p cos w. 

Substituting this value of x in the preceding equation, we 

obtain 

p 

p = p p COS W Or p =t; — r-4 > 

r r r r 1 + cos U 

which is the equation of the parabola referred to the focus by 
means of polar co-ordinates. 

XI. Equation of the ellipse referred to its own axes. 

Let (Fig. 8) the straight line AA' be equally divided in C, 
and let two points F, F' be taken on it on each side of C, and 
at an equal distance from it. Let also the curve line ABA'B' 
pass through A and A', and let the sum of the distances of each 
of its points from F and F' be equal to AA'. This curve is the 



20 PRINCIPLES OF ANALYTICAL. GEOMETRY. 

ellipse. C is its centre. AA ; ', represented by 2a, is called 
transverse axis, and BB', perpendicular to AA', and termi- 
nated by the curve, is called conjugate axis, and represented by 

CF 
26. The points F, F' are called foci, and the ratio ^ (< 1), 

, CF CF' . ;. 3 _ 
or its equivalent ratios — > is called the eccentricity, 

which is represented by e. Thus 

CF = CF' = ea. 

Taking now C for the origin of the orthogonal axes, and CAX, 
CBY for positive axes of abscissas and of ordinates, let M be 
any point of the curve, and CK, MK be the corresponding 
abscissa and ordinate x and y of M. Join M with F and F', 
MF, MF', called radius vectors, are represented by p and p'. 
From the triangles MFK, MF'K, right-angled in K, we have 

(r) \f = f + {x-eaf 
w \ P >* = tf + (x + eaf. 
Subtracting the first of these equations from the second we 
obtain the difference p' 2 — p 2 = 4 eax; i. e., 
(p r -f p) (p ; — p) = 4 eax. 
Now p' -f- p = 2 a ; hence p' — p = 2 ex ; hence also (p' -f p) 
+ ( P r — P ) = 2a + 2ex; but ( P ' + P ) + ( P ' — P ) = 2 P 'j 
therefore 

p' = a -f- ex and p /2 = a 2 + 2ea# -f eV. 

Substituting this last value in the second (r), we obtain 

a 2 -f e V = £/ 2 + a 2 -f- e V ; 
and consequently, 

(l)2, 2 =(l-6 2 )(a 2 -A 
which is the equation of the ellipse referred to the axes of the 
curve. Draw now from the foci the radius vectors FB, F'B, 

we shall have FB = FB = a. Now BC* = BF* — CF* and 
BC = b, CF = ea; therefore b 2 = a 2 — Jo 2 = a 2 (1 — e 2 ); 
hence 



PRINCIPLES OF ANALYTICAL GEOMETRY. 21 

(2) (1-^, = J 

This value substituted in the preceding formula (1) gives 
for the same equation of the ellipse 

(3) y 2 = 5(« 2 -^)- 

XII. Analysis and corresponding geometrical locus of the 

equation. 

Taking the square root of the last equation, we obtain 

b_ 
a 



y=±-S a 2 — x 2 



the geometrical locus of which referred to rectangular axes 
cannot give but the ellipse having its centre in the origin of 
the axes, 2a for the transverse and 26 for the conjugate axis, 
which coincide with the axes of reference. In fact, making 
x = o we obtain y == rfc 6, and making x = db a, y = o ; giv- 
ing to x different values from o to ± a, to each of these 
values correspond two values for y, one positive and one neg- 
ative, equal in length and diminishing when x increases. But 
no real value of y corresponds to any value of x greater than 
a. The curve is therefore re-entering and symmetrical. 
Moreover, if a > b, >/ a 2 — b 2 has a real linear value and 

V a 1 b 2 

less than a, thus < 1, calling e this ratio, and taking 

a 



on the transverse axis CF = CF' = s/a 2 — 6 2 , we shall have 

CF CF'_ 
a a 

Drawing now MF, MF' from any point M of the curve, 
whose co-ordinates x and y are CK, KM, we shall have 

MF = y 2 + (CK - CF) 2 = - 2 (a 2 -x 2 ) + (x — sTtf~Iv)\ 
MF' 2 *y + (CK + CF') 2 = § (a 2 -or) + (x + v^=T 2 ) 3 . 



22 PRINCIPLES OF ANALYTICAL GEOMETRY. 

b 2 



Now -2- (a 2 — x 2 ) = b 2 2 a? and (x=p \ / cf—b 2 ) 2 = x 2 =p 

Cb" CL 



2 X Vtf — b 2 + a 2 — b 2 . 
Substituting these values in the last members of the preceding 
equations, we deduce 

2/a 2 — b\ _^ ,_, -, 1 / xV a 2 — b 2 \2 

v a 2 ' x a 

^^ 
Hence MF = a — x = a — ex 

a 



^a 2 — b 2 
MF' = a -f x = a + ex ; 

(X 

and therefore MF + MF' = 2a ; i. e., 

The sum of the radius-vectors of any point of the curve is equal 
to the transverse axis, which is the characteristic property of the 
ellipse. 

It is plain from the equation of the ellipse that all the 
chords parallel to the conjugate axis, as MKM', are bisected 
by the transverse axis, and all the chords parallel to the trans- 
verse axis like MHM' are bisected by the conjugate axis, and 
all of them form right angles with the bisecting axes. This 
inference may be rendered more evident, relatively to the chords 
parallel to the transverse axis, by transforming the equation, 
into the following : 

a 



x = ± j ^b 2 — y\ 

> 

XIII. Parameter of the ellipse. 

For the ellipse, as for the parabola, the parameter is the 
double ordinate passing through the focus. Either of the foci 
will evidently give the same parameter. 

The equation of the ellipse which we have taken to analyze 
does not differ from the formula (1) XI. To obtain the para- 
meter it is enough to make x = CF = ea in the equation ; and 



PRINCIPLES OF ANALYTICAL GEOMETRY. 23 

taking the equation (1) for this purpose, we obtain (the 
parameter being represented by 2p) 

f = (1 -1 e 2 ) (a 2 — e 2 a 2 ) = a 2 (1 — e 2 ) 2 ; 
and from the formula (2) XI., 

p==a {l — e 2 ) = -; 

from which 2a : 2b = 26 : 2p. 

Thus, J7ie parameter, in the ellipse, is the third proportional 
after the transverse and the conjugate axis. 

XIV. Tangent and normal. 

Produce (Fig. 9) the radius vector F'M to N so that MX == 
MF. Join F with X, and draw from M, MET perpendicular 
to FX. This perpendicular is the tangent of the point M, for 
any other point of it is out of the curve. Let P be one of these 
points, join it with F, F r and with X. The equal right-angled 
triangles FME, XME give us EF = EX, hence also PF = 
PX. Xow from the triangle F'XP, F'P + PX >F'X; hence 
also FT + FP > F'X • but F'X = F'M + MF = 2a, there- 
fore F'P + FP > 2a ; the point P therefore is outside of the 
ellipse and PT touches the ellipse in M. 

Draw from M, MR perpendicular to the tangent. This per- 
pendicular is called the normal of the point M. Since MR is 
parallel to XF, the angles RMF, MFX are equal, as also the 
angles F'MR, MXF; but MFX = MXF, therefore RMF = 
F'MR ; i. e., The normal of any point of the ellipse bisects the 
angle formed by the radius vectors of that point. 

It follows then that F'MP = FMT; i. e., 

The radius-vectors of any point of the ellipse form equal 
angles with the tangent. 

XV. Diameters of the ellipse — Conjugate diameters. 

A. straight line passing through the centre of the ellipse and 
terminated on both sides by the periphery, is called diameter. 



24 PRINCIPLES OF ANALYTICAL GEOMETRY. 

Let now (Fig. 10) DCD' be one of these diameters passing 
through the middle point O of the chord MM', which we shall 
represent by 2c. Draw from M, O, M' the perpendiculars 
MK, OH, M'K' to the transverse axis AA', and, from O and 
M', the perpendiculars ON, M'N' to MK, OH. Call x„ y, the 
co-ordinates CH, HO of the middle point O of the chord re- 
ferred to the orthogonal axes A' AX, B'BY, and let (3 be the 
angle which MM' forms with the positive axis of abscissas. 
From the equation (3) XI. of the ellipse referred to these same 
axes, we have 

b 2 






2 X 



therefore KM* = b 2 — -- CK* K'M' 2 =b 2 —- 2 CK /2 : but 

a 2 a 2 

KM=HO + NM = 2/,+ csin/?,CK = CH— ON=a?,+ ccos/3. 

K'M'=HO— OW=y — csin/3,CK'=CH+N'M'=#,— ccos/3. 

Making a substitution of these values in the preceding 

equations, we obtain 

b 2 
(y,+o sin f3f = b 2 2 (x, + c cos /3) 2 

7 2 

{y, — c sin (3) 2 = b 2 — - 2 (x, — c cos /3) 2 ; 
and taking the second of these formulas from the first, 

4 y, c ' sin (3 — — 4 — 2 x, c * cos p ; 

b 2 cot(3 
hence also (2) y, = -% — x,. 

Now the angle /3 does not change for any chord parallel to 
MM', hence the last equation would be obtained in equal 
manner for the co-ordinates of the middle point of any chord 
parallel to MM' ; hence the same equation represents the geo- 
metrical locus passing through the middle points of a system 
of parallel chords. But the geometrical locus represented by 



PRINCIPLES OF ANALYTICAL GEOMETRY. 25 

the equation is (III. 1st) a straight line passing through the' 
origin of the axes, hence all the chords parallel to MM' are 
bisected by the diameter DD' ; and to bisect any system of 
parallel chords in the ellipse, it suffices to draw a diameter 
from the middle point of any one of them. 

It follows, from what precedes, that the diameter EE' par- 
allel to MM' is also bisected by DD', and, as we shall see in 
the next article, EE' bisects in its turn all the chords parallel 
to DD'. These diameters, each one of which bisects the chords 
parallel to the other, are called conjugate diameters. 

Calling now a, the angle which DCD' forms with the posi- 
tive axis CX, the equation of D'CD referred to the axes A'X, 
B'Y is (III.) y = tgax ; but the equation of the same line, as 

we have seen above, is also y = - t — x, hence tga = — 



a" 



b 2 cot jS . +1 

— 2 — > an( i consequently 



<r 



b 2 
(3) tgatgP= . 



a 2 



Therefore, the condition to be verified, in order that two 
diameters be conjugate diameters, is that the product of the 
tangents of their respective angles with the positive axis of 
abscissas be equal to the negative quotient of the square of 
the conjugate semiaxis divided by the square of the transverse 
semiaxis. 

To determine the length of the conjugate semidiameters 
CD = a', CE = b r , observe that the co-ordinates x and y of 
the extremity D of CD are respectively equal to a' cos a, 
a' sin a. Now D is one of the points of the ellipse represented 
by the preceding equation (1); therefore, the co-ordinates of 
this point substituted in (1) fulfil that equation, i. e., a n sin 2 a 



b 2 

= b 2 ^ a' 2 cos 2 a, or 

a 2 



a n {a 2 sin 2 a -f b 2 cos 2 a) = b 2 a 2 . 
In like manner the co-ordinates x and y, of the extremity 



26 PRINCIPLES OF ANALYTICAL GEOMETRY. 

E of CE, being respectively represented by b f cos (3 and b 1 sin [3, 
substituted in the same equation (1), give us 

b' 2 (a 2 sin 2 (3 + b 2 cos 2 /S) = b 2 a 2 . 

Hence, from this and from the preceding formula, we infer 

a 2 6 2 



a' 2 = 



(4) i 



a 2 sin 2 a -f b 2 cos 2 a 



2*2 

7 ,„ a" er 

b = 



a 2 sin 2 /3 + 6 2 cos 2 /3. 



XVI. Equation of the ellipse referred to conjugate diameters. 

Let us now refer the curve to the conjugate diameters, taking 
D'CX' for axis of abscissas and E'CY' for axis of ordinates. 
Representing by x f y y' the co-ordinates of the curve referred 
to this system of axes, Ave shall have from the formulas (1) 

ii., 

x = x r cos a 4- y f cos /3, y = x r sin a -f y f sin (3. 

These values substituted in the equation (1) of the preceding 
paragraph, first reduced to the following form, 



x 



r_ 



give /cos 2 a sin 2 a > ,, ,cos 2 /3 sin 2 ^> ,, 

_ /cos a cos /3 sin a sin (3\ 

+ 2 ( ^ + gr— ) ? » = If 

Now from the formula (3) of the preceding paragraph XV. 

b 2 
we infer sin a sin (3 — cos a cos /?, and this value sub- 

ar ' 

stituted in the last equation destroys its third term. With 

regard to the first and second term, which are equivalent to 

a 2 sin 2 a + i 2 cos 2 a a 2 sin 2 (3 + b 2 cos 2 [3 ,, , , „ , „ 

7,-77, -, — t- w , the last formulas ot 

or 6 W a 2 b 1 

the preceding paragraph change them into -725 p: 2 ; hence the 



PRINCIPLES OF ANALYTICAL GEOMETRY. 27 

above equation assumes the form 

(2) -, + ^1, 

altogether similar to the preceding (1) which represents the 
ellipse referred to the axes; and as from the nature of that 
equation we infer that all the chords parallel to one of the 
axes is necessarily bisected by the other axis, so also it fol- 
lows from the last equation that all the chords parallel to one 
of the conjugate diameters are bisected by the other diameter. 

XVII. Polar equation of the ellipse. 

We have found (XI.) that p -f p r = 2a and p f = a -f- ex. 
Taking the focus F (Fig. 9) for pole and AA r for polar axis, 
and calling w the angle MFA. Since x = CK = CF + FK 
and (XI.) CF = ea, FK = p cos w, we have also x = ea -f- 
p cos w, but from the above equations p = 2a — p' = 2a — a — 
ex = a — ex, therefore 

p = a — e 2 a — e p cos w ; 

hence (1) p = — - - ■ 

v } v 1 + e cos u ' 

b 2 

or (XIII.) on account of a (1 — e 2 ) = p = — , 

(2) P b ' 

^' ' 1 + e cos w " a (1+ e cos w) 

Each of these equations, having no other variables but u and 
p, represents the ellipse referred to the polar co-ordinates. 

XVIII. Theorems concerning the axes and conjugate diameters. 

From the first of the formulas, (4) XV., we obtain 
a' 2 a 2 sin 2 a + a! 2 b 2 cos 2 a = a 2 b 2 = a 2 b 2 (sin 2 a + cos 2 a), 
which, divided by cos 2 a, and resolved with regard to tg 2 a, 
gives 

2 _ b 2 (a 2 — a f2 ) 
9 a ~ a 2 {a ,2 — b 2 )' 



28 PRINCIPLES OF ANALYTICAL GEOMETRY. 

In like manner we obtain from the second formula 

_ b 2 {a 2 -b> 2 ) 
9 P-aWZ — b 2 )' 
and from these two 

. , ,,, & a i — a 2 a f 2 —a 2 b ' 2 + a' 2 b' 2 
r * * * " a 4 * a! 2 V 2 — IW^^W'fP ; 

6 4 
but from the equation (3) XV. t(f a tg 2 (3 = ~ i9 hence 

a 

a ± a 2 a /2 a 2£/2 _^_ a /2^/2 

^72 _ &/2fr 2 _ dW +tf = *> and conse q u ently 

a * _ a 2 ( a , a + 6 «j = tf — b 2 {b' 2 + a' 2 ) ; 

from which a 4 — 6 4 = (a 2 — b 2 ) (a /2 + &' 2 ). Now a 4 — 6 4 = 
(a 2 + 6 2 )(a 2 — 6 2 ), therefore 

a' 2 + &' 2 = a 2 + 6 2 , 
and 

4a' 2 + 46' 2 = 4a 2 + 46 2 ; 

i. e., in the ellipse, The sum of the squares of the axes is equal 
to the sum of the squares of the conjugate diameters. 
From the same formulas, (4) XV., we have 

a 2 6 2 o a 2 b 2 

a 2 sin 2 a + b 2 cos 2 a == — w . a 2 sin 2 (3 -f b 2 cos 2 /3 — -^i 

a b 

hence 

a 4 sin 2 a sin 2 /3 -f a 2 b 2 sin 2 a cos 2 /3 -f a 2 b 2 cos 2 a sin 2 (3 

a A ¥ 



+ 6 4 cos 2 a cos 2 /3 = 



/2Z,/2* 



a f2 b 



Now from the equation, (3) XV., we have 

a 
which squared gives 

tg 2 *tg 2 (3+2tg*tgf3- 2 + ^ = 9 

a a 

and also 



PRINCIPLES OF ANALYTICAL GEOMETRY. 29 

a 4 sin 2 a sin 2 {3 -f 2a 2 6 2 sin a sin f& cos a cos /3 + b 4 cos 2 a cos 2 j3 = ; 

hence 

a 4 sin 2 a sin 2 /3 + b 4 cos 2 a cos 2 j3 == — 2a 2 6 2 sin a sin /3 cos a cos p\ 

This value substituted in the preceding equation gives us 
a 2 b 2 sin 2 a cos 2 /3 + a 2 b 2 cos 2 a sin 2 /3 — 2a 2 6 2 sin a sinp 1 cos a cos ^ 

a 4 & 4 
~ a ,2 b n ' 

Suppressing the common factor a 2 b 2 , and taking the square 
root, we obtain the formula, 

• o o • a '° 

sin p cos a — cos ,o sin a = -r-.-.. 

a' 'b 
Now (Trig., p. 253, h") sin (3 cos a — cos £ sin a == sin (/3 — a) ; 
hence 

a • 5 = a' * V sin (/3 — a). 

Observe that a * 6 represents the area of the rectangle con- 
structed on the semi-axes, and a' ' b f sin (f3 — a) the rectangular 
area of the parallelogram constructed on the conjugate semi- 
diameters ; therefore, Tlie parallelogram constructed with the 
conjugate diameters is equivalent to the rectangle of the axes. 

XIX. Equation of the hyperbola referred to its axes. 

Take (Fig. 11) the point C on the indefinite straight line XX' 
and on each side of C two segments CA == CA', CF (>CA) 
= CF'. Let also a curve MAM' pass through A, and a corre- 
sponding one through A', and let the difference of the distances 
of each point of the curve, on either side, from F and F' be always 
the same and equal to AA'. This double curve is the hyper- 
bola, which has C for centre and AA', represented by 2a, for 
transverse axis. The points F, F' are called the foci, and the 
points A, A' of the curve met by the axis are called the vertices. 
Any straight line passing through C and terminated on both 

CF CF CF' 

sides by the curve is called a diameter, -^-t = — = is 

CA a a 



30 PRINCIPLES OF ANALYTICAL GEOMETRY. 

called the eccentricity, which we represent by e, and since 
CF>CA, 

CF = CF'_ 
a a 

Draw now from the centre C, YY' perpendicular to AA', and 
taking XX' for axis of the abscissas and YY' for axis of 
ordinates, let x, y be the co-ordinates CK, MK of any point M 
of the curve. Since from the above equality Ave have 

CF = CF' = ae, 

calling p, p' the distances MF, MF' of the foci from the point 
M, which distances are called radius-vectors, the right-angled 
triangles MKF, MKF' give 



« f ■ 



p 2 = y 2 -f (x — ae) 2 



f = f + (x + ae) 2 . 
Taking the first of these formulas from the second, we have 

(p' + p)(? f — p) = ^x', 
but p' — p == 2a ; hence p' -f p = 2ex ; hence also adding to or 
subtracting from each other these last two equations, 

(r') p' = a + ex, p = ex — a. 
Substituting in the second (r) the value of p' last obtained, 
that formula will become a 2 -f- e 2 ^ 2 = y 2 + % 2 + a 2 e 2 , from 
which 

(1) tf = (<?-l){,?-a% 

the equation of the hyperbola referred to the rectangular axes 
XX', Y Y'. To eliminate the eccentricity from this equation, call 
c the distance CF; we shall then have c = ae, and consequently 

2 2 

e = — , and e 2 — 1 = 5 — . Let now b represent the mean 

a a 2 

geometrical proportional between c -{- a, c — a, we sha 1 ! have 

b 2 = c 2 — a 2 , and therefore 

b 2 
(2) e 2 — l = - 2 . 

This value changes the equation (1) into 



PRINCIPLES OF ANALYTICAL GEOMETRY. 31 

(3) f = ^- a % 

To keep the analogy with the ellipse, taking on each side of 
YY' from the centre CB = CB' = b, the segment BB' of 
YY'is called the conjugate axis of the hyperbola. 

» 

XX. Analysis and corresponding geometrical locus of the 

equation. 

From the last equation (3) Ave obtain the following : 

from the analysis of which we infer, 1st, that no real value 
of y corresponds to the values of x, either positive or negative, 
from to a. 2d. That the ordinate y = corresponds to 
the abscissas x — a, x = — a. 3d. Two real orclinates, one 
positive and one negative, but of equal numerical value, cor 
respond to the values, either positive or negative, of x when 
its numerical value is greater than a, and the numerical value 
of y increases indefinitely with that of x. Referring, therefore, 
the geometrical locus represented by the above equation to the 
orthogonal axes XX', YY', we find it to cut the axis of 
abscissas at the distances a, — a from the origin, and recede 
thence from the axis of ordinates divided into four symmetri- 
cal branches, two above and two below the axis of abscissas. 

Taking on the axis XX' two points F, F'' equidistant from 
the centre, the distance being CF == CF' = *S a 2 + b 2 (>a), 

CF CF' 

so that — = — - >1, and representing as usual by e the 

(Jj CI/ 

CF 

ratio , CF = CF r = ae • from the right-angled triangles 

a 

MFK, MF r K with a process similar to that followed for the 

ellipse^ we shall find for the positive values of both MF, MF', 

MF = ex — a, MF' = ex + a. 



32 PRINCIPLES OF ANALYTICAL GEOMETRY. 

Hence MF' — MF = 2a, which is the characteristic property 
of the hyperbola. It may be well to remark that the formula 
from which we obtain the value of MF gives indifferently a — 
ex and ex — a, but as in the case of the ellipse e being <] 1 
and x never greater than a, the positive value of MF can be 
given only by a — ex, so in the case of the hyperbola for 
which e > 1 and x never less than a the positive value of MF 
is only obtained by ex — a. 

XXI. Parameter of the hyperbola. 

The parameter of the hyperbola, like that of the preceding 
curves, is the double ordinate which passes through the focus ; 
2p representing the parameter, p is the ordinate corresponding 
to the abscissa CF = ae. Substituting, therefore, in the equa- 
tion (1) XIX., a,e for x, the same equation gives us 

2 2/2 1 \2 

pr = a (er — 1) ; 

hence from the equation (2) of the same paragraph 

b 2 b 2 
p= a- -2= —, 

a 2 a 

as for the ellipse. Therefore, since from this last equation, 

2a :2b: :2b: 2p. 

For the hyperbola, as for the ellipse, the parameter is the 
third proportional to the transverse and the conjugate axis. 

XXII. Tangent and, normal. 

Let M (Fig. 12) be any point of the hyperbola, and MF, 
MF' its distances from the foci or radius-vectors. Take on 
MF', MN = MF, and connecting F with X, draw MT per- 
pendicular to XF. MT is the tangent of the point M of the 
curve. Taking, in fact, on either side of M a point P on 
TMT', and connecting it with F, F' and X, we shall have 
PF = PX and PF' < PX + XF' ; hence PF' — PF < XF'. 
Now, NF' = MF' — MF = 2a j hence PF r — PF < 2a, P 



PRINCIPLES OF ANALYTICAL GEOMETRY. 33 

therefore is not one of the points of the curve. The same 
result would be obtained for any point of MT taken on the 
side of T', hence none of the points of TT', except M, is on the 
curve. TT' is, moreover, altogether on the convex side of the 
curve, a condition to be fulfilled in order that TT' be a tan- 
gent. Taking, in fact, any point between the transverse axis 
and the curve, on the concave side, and drawing from it a 
perpendicular to the axis produced on the other side, until it 
reaches the curve, connecting then the point of the curve met 
by this perpendicular and the point from which the perpen- 
dicular is drawn with the foci, we shall find the difference of 
the distances of the last point greater than that of the radius- 
vectors of the point of the curve ; hence the same difference is 
> 2a. Now the difference of the distances of any point of TT' 
on either side of M from the foci is < 2a ; hence all these 
points and consequently the whole straight line TT' is on the 
convex side of the curve. 

From the equal triangles NEM, FEM we infer the equal- 
ity of the angles formed by the tangent with the radius- 
vectors. To draw, therefore, from any point of the hyperbola 
a tangent, divide by a straight line the angle formed by the 
radius-vectors of that point into two equal parts. It is easy to 
see that producing FM to X' and drawing from M, MB, per- 
pendicular to the tangent, (MR is called the normal of M,) 
the normal also divides into two equal parts the angle formed 
by the radius- vectors ; i. e., 

The angles formed by the radius-vectors of any point of the 
hyperbola are bisected, one by the tangent and the other by the 
normal. 

XXIII. Asymptotes of the hyperbola. 

The asymptote is a straight line approaching indefinitely to 
a curvilinear branch or branches without ever reaching them. 
The hyperbola admits of two of these asymptotes, which are 
the indefinitely produced diagonals UX', VY' (Fig. 13) of 



34 PRINCIPLES OF ANALYTICAL GEOMETRY. 

the rectangle DED'E', constructed upon the axes 2a, 2b of 
the curve. 

Let, in fact, CK be the abscissa x corresponding to KM, 
the ordinate y of the curve, and to KL the ordinate y 9 , of V Y, 
the prolonged diagonal E'D of the rectangle. From the equa- 
tions of the hyperbola referred to its own axes, and of the 
straight line referred to the same axes, we shall have at once, 

a \/ a 

and consequently, 

b , /-^ ^ ab 

y, — y — — (x — v or — or) = 



a x + ^x 2 — a 2 . 

It is plain from the first two equations, that whatever be x 
the corresponding y, is greater than y, and from the last equa- 
tion, that the greater is x the smaller is the difference y, — y. 
Hence CY', even indefinitely prolonged, is altogether outside 
of the branch AM of the curve, but approaching to it more 
and more, the m^ore the branch of the curve and the diagonal 
recede from the centre. The same demonstration is applicable 
to the other branches. Hence the hyperbola admits of two 
asymptotes VY', TJX', each approaching in opposite direc- 
tions to two of the branches of the curve. 

XXIV. Equation of the hyperbola referred to the asymptotes. 

Representing by a and — a the equal angles which the 
asymptotes CY ; , CX r form with the axis CX, and represent- 
ing by a/, y ! the co-ordinates of any point M of the curve 
referred to them, as x, y represent the co-ordinates of the same 
point referred to the axes of the curve ; we shall first obtain 
from the general formulas (1) II., 

x = x' cos a -f y' cos a, y = y' sin a — x' sin a, 

these values substituted in the equation (3) XIX., which can 
easily be reduced to the following : 



PRINCIPLES OF ANALYTICAL GEOMETRY. 35 

2 2 

__2/_ = i 
a 2 b 2 ' 

will give 

.cos 2 ^ sin 2 a^ / cos 2 a sin 2 a^ , 2 

.cos 2 a . sin 2 a> 



2C-^- + ^V->y=i. 



Now — ^ — = to 2 a = -^ ; hence 
cos 2 a <r 



sin 2 a cos 2 a 



6 2 a 2 ' 

b 2 
and, consequently, sin 2 a = — cos 2 a ; hence 

sin 2 a + cos 2 a = cos 2 a (l + — J = 5— (a 2 + & 2 ). 

v a 2/ a 2 v y 

2 -j 

Now sin 2 a + cos 2 a = 1 : hence — — = -5 - , and therefore 

<r a + 6" 

sin 2 a cos 2 a 2 

+ 



6 2 'a 2 a 2 + 6 2 
Thus the preceding formula becomes 

Ax r y r 



a 2 +b 2 
or, 



1, 



, , a 2 + b~ 
x'y f = 



4 ' 

which is the equation of the hyperbola referred to the asymp- 
totes. Now a 2 + b 2 or c 2 is (XIX.) the square of the distance 

of each focus from the centre; therefore x'y r =■ I — Y; i. e., in the 

equation of the hyperbola referred to the asymptotes the pro- 
duct of the co-ordinates is constant and equal to the square of 
one-half the distance of each focus from the centre. If the 
axes 2a, 26 be equal, in which case the hyperbola is called 
equilateral, the angle formed by the asymptotes is a right 



36 PRINCIPLES OF ANALYTICAL GEOMETRY. 

angle, and the equation of the curve referred to them becomes 

9 2 

XXV. Polar equation of the hyperbola. 

From the characteristic property of the hyperbola expressed 
by the equation p' — p == 2a, and from the first equation (r r ) 
XIX., we obtain 

p = p' — 2 a = ex — a. 

Calling now w the angle AFM (Fig. 11) formed by the 
radius-vector of any point M of the curve with the axis XX', 
which we take for polar axis, we shall have 

x = CF + FK = ae — p cos u ; 
hence 

p = e (ae — p cos u) — a = a (e 2 — 1) — e p cos w ; 

, „ a{# — 1) 
therefore, p = - — t -. 

1+6 COS GJ 

Now (XXI.) a (e 2 — 1) is the semiparameter p of the 
hyperbola; hence, 

„ P_ 

P = i — ; ' 

1 + e cos u 

which is the polar equation. 

XX VI. General polar equation. 

Comparing this equation with the polar equations of the 

parabola, X., and with that of the ellipse, (2) XVII., we find 

that the same formula, 

P 

1 + e cos w 

represents the ellipse, the parabola and the hyperbola, accord- 
ing as the eccentricity 6is<l, = lor>l;j9 representing 
the semiparameter of each curve: the pole is taken in the 
focus or in one of the foci, and the angle w commences on the 
side of the nearest vertex to the pole. ' 



PRINCIPLES OF ANALYTICAL GEOMETRY. 37 

XXVII. Equation of the cycloid. 

We call cycloid the curve BAB' (Fig. 14) produced on the 
plane BEAB' by a point B of the circular periphery BNE, 
while the circle, touching constantly the straight line BB', 
revolves until the point B conies again into contact with the 
straight line in B'. The rolling circle 3NE is called the 
generating circle, whose diameter we shall represent by 2c. 
The straight line BB', which is = 2c ir, is called the base of the 
curve ; and AA' perpendicular to BB' and passing through the 
middle point A' of the base, is called the axis of the cycloid, 
and the extremity A of this axis, vertex of the curve. It is 
plain that the axis is equal to the diameter 2c of the gener- 
ating circle. 

Placing the origin of orthogonal axes, to which the cycloid 
is to be referred, in the vertex A let the axis AX of abscissas 
coincide with the axis of the curve, and let AY parallel to 
BB' be the axis of ordinates. Let also x, y be the co-ordinates 
AK, KM of the point M of the cycloid corresponding to the 
position DMD' of the generating circle. The diameter DD', 
whose extremity D' is the point of contact with the base, is 
necessarily perpendicular to the same base, and consequently 
parallel to AX. 

Call now a the arc of the circle having unity for radius, 
and measuring the same angle measured by MD the supple- 
ment of MD'. From the genesis of the cycloid we have 

BA' = DMD', BD' = MD'; 
hence 

HK = BA' — BD' = DM = ca; 

hence, also, 

X — DH = C V ' sin a = c (1 — COS a), 

y = KH + HM = ca + c sin a = c (a + sin a). 

Now MH 2 = HD' • HD = (2c — x) x = 2cx — x 2 or MH = 

*S2cx — a?. But MH — c sin «, therefore sin a = . 



38 PRINCIPLES OF ANALYTICAL GEOMETRY. 

hence from the second of the last equations 



y = c arc (sm = [ ) + V lex — x\ 

which is the equation of the cycloid referred to the above-men- 
tioned axes. 



XXVIII. Rectilinear and polar co-ordinates of points in • 

space. 

Conceive three planes XAZ, ZAY, YAX passing through 
the same point A (Fig. 15) ; AX, AY, AZ being their mutual 
intersections. Let now M be a point in space, i. e., placed 
somewhere out of each of the three planes. The position of 
M relatively to the three planes or to A is in this case deter- 
mined by means of three co-ordinates, as follows : Draw from 
M, MH parallel to AZ, called also axis Z, until it reaches the 
plane XAY in H, MK' parallel to AY, or axis Y, until it 
reaches the plane ZAX in K' and MD' parallel to AX, or 
axis X, until it reaches the plane YAZ in D f . These three 
parallels determine the position of M in space relatively to the 
three planes ; for the same three parallels cannot simultaneously 
belong to any other point. Now, the two parallels MK', MH 
determine the position of a plane parallel to ZAY, which pro- 
duced, will cut the axis X, and let K be the point of inter- 
section. In like manner the parallels MK r , MD' and the 
parallels MD', MH determine the positions of planes respec- 
tively parallel to XAY and XAZ, and each of them produced 
will cut the axes Z and Y, the first say in A', the second in 
D. Connecting now K with K' and H, A' with K' and D f , 
and D with T> f and H, we obtain a parallelopipedon, and con- 
sequently MK' = HK and MD' = AK. Therefore to deter- 
mine the position of M relatively to the three planes, it is 
enough to draw the parallel MH, or co-ordinate z, and from 
H, HK or co-ordinate y parallel to AY, and take in connec- 
tion with them the segment AK or co-ordinate x on the axis 



PEINCIPLES OF ANALYTICAL GEOMETRY. 39 

AX. These three co-ordinates determine the position of M 
relatively to the origin A of the axes. These co-ordinates 
will be regarded as positive or as negative according as their 
directions are toward X, Y, and Z, or toward the opposite 
sides. Varying the values of these co-ordinates x, y, z from 
— oo to -f- oo we can evidently obtain the position of any 
point in space. The axes AX, AY, AZ are commonly taken 
at right angles, in which case the co-ordinates are said to be 
orthogonal. 

In this supposition join A with M, A being taken as pole, 
AM is the radius-vector p of M. Call & the angle which p 
makes with AZ, and w the angle which AH makes with AX, 
which is the angle formed by the planes ZAH, ZAX. These 
three elements p, d, w, determine the position of the point M in 
space relatively to the centre A, and to the axes, for three 
given values of these elements cannot belong simultaneously 
to more than one point, and taking p from to + oo, & from 
0° to 180° or <*, w from 0° to 360° or 2*, the position of every 
point in space can be determined by means of them. These 
are the polar co-ordinates of points in space. We may from 
these co-ordinates obtain the rectilinear co-ordinates of the 
same points, or vice versa. In fact, we have from the right- 
angled triangles AKH, AHM, 

AK = AH cos w = p cos (90° — 0) cos w = p sin cos w, 

HK = AH sin w = p sin & sin w, 

MH = p cos Q. 

Now AK, HK, MH are the rectilinear co-ordinates x, y, z of 
M; hence 

x = p sin & cos w, y == p sin & sin w, z — p cos 6. 

From the same right-angled triangles, 

.2 2 2 2 2 



AM = AH + MH = AK + KH + MH , 

MH 
MH=AM cos MAZ; hence cos MAZ 



^AK 2 +KM 2 +MH 2 , 

HK : AK : : sin HAK : cos HAK : hence to HAK = S. 

; * ' AK 



40 PRINCIPLES OF ANALYTICAL GEOMETRY. 

Substituting the corresponding values, the same equations 
become 

p = vx 2 4. y 2 -f z* cos 6 = / — 9 „ — —„ tq a = ■£-. 
r ^ + 2/ 2 + z 2 a? 

XXIX. Equation of the plane. 

Let (Fig.16) AX, AY,AZ be three orthogonal axes, and BCD 
an indefinite plane in space, which meets the axis AZ in C and 
cuts the planes ZAX, ZAY along the straight lines BCF, 
DCI. Let MH, HK, KA be the co-ordinates z, y, x of any 
point M of the plane. Since the plane determined by MH, 
HK is parallel to ZAY, the intersections UF, TI' of MHK 
produced, with the plane of the axis AX, AY, and with the 
given plane BCD, are respectively parallel to YI, DI. In like 
manner the intersection KV of the same plane MHK with the 
plane of the axes AZ, AX is parallel to AZ. Hence UKY 
= YAZ = 90° and UFT = YID. Call q the segment AC of 
the axis AZ between the origin and the point met by the plane 
in space, m the tangent of the angle XFB, and n the tangent 
of the angle YID. Concerning the straight line FT, referred 
to the axes KU, KV, we shall have (3) III., MH = n ' KH 
+ KE ; i. e., 

z = ny + KE ; 

but with regard to F B referred to the axes AX, AZ w^e have 

KE = mx -\- q ) 
hence 

z — mx -f ny -f q, 
an equation between the constants m, n, q and the co-ordinates 
x, y, z of the point M of the plane ; but M is any point of the 
given plane; hence the last equation is the equation of the 
plane, and for any values taken at pleasure for x and y, we 
may obtain through it the value of the third co-ordinate z. 



PRINCIPLES OF ANALYTICAL GEOMETRY. 41 

Should the plane pass through the origin of the axes, then q 
= 0, and the equation becomes in this case 

z = mx -f ny. 

Representing m by — , n by — — r , and q by ^-, the 

last and the preceding equations are easily changed into 
Ax + By + Cz + D = 0, 
Ax + -By + Cz = • 
the first of which represents any plane at all ; the second, any 
plane passing through the origin of the axes. 

XXX. Equations of the straight line in space. 

Let (Fig. 17) any straight line RR' in space be referred to 
the orthogonal axes AX, AY, AZ. Draw from any point M 
of RR', MH perpendicular to the plane of the axes AX, AY, 
and MN perpendicular to the plane of the axes AY, AZ, and let 
PP' be the intersection between the plane of the axes AX, AY 
and the plane determined by the line RR' in space and the 
perpendicular MH. Let also QQ' be the intersection between 
the plane of the axes AZ, AY and the plane determined by 
the line RR' in space and the perpendicular MX. These 
two intersections PP', QQ' are called projections of the 
straight line in space, the first on the plane XAY, the second 
on the plane ZAY. Let now AK, KH, HM be the co-ordi- 
nates x y y, z of the point M. The first two x, y belong also to 
the point H of the projection PP r referred to the axes AX, AY, 
and the two y, z belong also to the point N of the projection 
QQ' referred to the axes AZ, AY. Now, (3) III., let 

y = ax + b, y = a'z + b' 

be the equations of the projections each referred to the axes of 
its own plane. By means of them the straight line RR r in 
space may be also represented. For taking the value of any of 
the three co-ordinates, by means of the two equations we 



42 PRINCIPLES OF ANALYTICAL GEOMETRY. 

obtain the other two, and all the three co-ordinates belong to 
one of the points of the line in space. 

To conclude these sketches of analytical geometry, we may 
remark that as a plane surface in space, so also a curved sur- 
face may be referred to the orthogonal axes. Its equation, 
however, would be found of a degree higher than the first. 
And as a straight line in space can be referred to the same 
axes and be represented by the equations of the projections of 
the line on two of the planes formed by the same axes, so 
likewise a curved line in space can be referred to these axes, 
and represented by the equations of the projections of the curve 
in space upon two of the planes of the axes. 



PEINCIPLES OF 

INFINITESIMAL CALCULUS. 



PART I. 

DIFFERENTIAL CALCULUS. 



I. Infinitesimal quantities; different orders and expressions of 

the same. 

We call that quantity infinitesimal which is conceived to be 
less than any given quantity of the same kind, however small. 

Representing by a an infinitesimal quantity, which we shall 
call of the first order, the powers a 2 , a 3 , a 4 , . . . a. 71 of the same 
quantity will be infinitesimals of the second, of the third . . . 
of the ?itli order; inasmuch as in the series of these quanti- 
ties each must be regarded as infinitely less than the preceding 
and infinitely greater than the following. 

Thus if any quantity (3 divided by the infinitesimal a gives 
the finite quotient x, (3 must be regarded as an infinitesimal of 
the first order ; and if 



(3 y 5 



CO 



u a 2 a? ~ " ' a n ~~ *' 

y, <5, . . . w must be regarded as infinitesimals of the 2d, 3d, 

. . . nth orders. 

Hence the infinitesimals of different orders can be expressed 

as follows : 

(3 = xoij y = xa 2 , 8 = xa 3 , ... to = xa™ ; 

that is, 

43 



44 PRINCIPLES OF INFINITESIMAL CALCULUS. 

The product of a finite quantity by an infinitesimal of any 
order is an infinitesimal quantity of the same order. 

We must add to these preliminaries the following principle, 
generally admitted in the analysis of infinitesimal calculus, 
and found to be correct in all its physical applications, i. e., 
Infinitesimal quantities disappear when compared with finite 
quantities, or when compared ivith infinitesimal quantities of a 
lower order, which comes to the same as to say, two finite 
quantities which differ from each other by an infinitesimal 
one, or two infinitesimal quantities which differ from one 
another by an infinitesimal of a higher order, are or may be 
considered as identical. In fact, it follows from the definition 
of the infinitesimal quantity, that the difference in both cases 
is less than any quantity which can be assigned. 

IT. Functions. 

A quantity is called constant or variable according as it has 
a fixed or variable value. 

Variable quantities may and do frequently depend on each 
other, and then they are said to be functions of one another. 
Thus, for example, if by changing the value of the quantity x 
the value of another quantity y is also varied ; y is called a 
function of x, and vice versa. That y is a function of x is 
expressed by the equation 

In this equation x is called an independent variable, inasmuch 
as we make y depend on any value arbitrarily given to x. If 
Ave should make x vary according to the arbitrary values given 
to y, then y would be the independent variable, in which case 
the dependence of x on y would be expressed by the equation 

x=F(y), 
the capital letter F being used instead of /, to signify the dif- 
ferent form of the function, as <p, % . . . would be used for other 
functions differing from the preceding. 



DIFFERENTIAL CALCULUS. 45 

These are called explicit functions, to distinguish them from 
the implicit, in which the function is not immediately given 
by the independent variable ; as, for instance, in the equation 

y 2 — 2xy — a = 0, 
in which y is a function of x, but not immediately given by x. 
If the equation be resolved, we then have 

y = x d= v a + a? ; 
i. e., y given immediately by x ; and representing x ± *S a + x 2 
by Fie, y = Fee is the explicit function of x deduced from the 
implicit one by means of the resolution of the given equation. 

III. Differentials. 

Let b 

y=ax,y = -- 

be two equations between x and y, which also represent 
two different functions of x. By increasing x, the function 
y increases in the first and decreases in the second equation, 
and vice versa, whatever the increase of x may be. In 
every case, however, an infinitesimal change of x is neces- 
sarily attended by a change of y equally infinitesimal. Repre- 
senting thus by dx, dy the infinitesimal variations of x and of 
y, we deduce from the preceding equations the two following : 

y ± dy = a (% nfc dx), y ± dy = —r—f> 

and supposing, as is often done, that dx represents the increase 
as well as the diminution of the variable x, and dy the increase 
and the diminution of the function ; the above equations are 
more simply written as follows : 

y -f dy = a (x -f dx), y -f dy = ^; 

(fa is called the differential of a?, and d?/ the differential of y or 
of the function of x. From the last and the given equations 
it is easy to see that 



46 PRINCIPLES OF INFINITESIMAL CALCULUS. 

b b 



x -f dx x 
And generally supposing the given equation to be y = / (x), 



dy = a (x -f cfc) — ax, or c?y = 

merally 
we shall obtain 

dy = f{x + dx) —f(x); 

that is, 

The differential of any function of x is egi£«£ to the difference 
between the first and the second state of the function, the second 
state being the result of an infinitesimal change of x in the given 
function. 

Differential calculus has for its object to determine the dif- 
ferentials of given functions. But before we proceed to give 
the principal and most common rules of this calculus, the fol- 
lowing theorems require to be demonstrated. 

IV. Preliminary theorems. 

Let n be a positive whole number increasing indefinitely 
and having for its limit infinity, i. e., a value superior to any as- 
signable value, and take the binomial (l H ) n , in which, when 

n has reached its limit, — is necessarily infinitesimal. We 

' n J 

know from algebra (see Treat. § 69) that 

(1 + I)- - 1 + ••. 1 + .«*£!>. \ + nin-mn- Z) L 
v n J n 2 n 2 2 • 3 n 3 

, . n(n — l)(n — 2) (n — (n — 1)) 1 

2-3 n n 11 ' 

The second member of this equation can be easily transformed 
into the following : 

2 + 1(1-1) + JL(i--) (i--) + ---- 

2 v n> 2 * 3 v n /v n' 

1 /-i 1\ a 2 x H n — l x 
+ 2T3T77^(l--)(l--)—(l ^), 



DIFFERENTIAL CALCULUS. 47 

in which all the factors 1 , 1 , * * * v l , are 

n n n 

positive, and their number and values increase with n ; hence 

also (1 + — ) n increases in value with n. But whatever n may 

* 
be, each one of the same factors is less than 1 ; hence 

t (i - -) < i A (I -- -) (i - -) < A' et0 -> 

2 ^ w y 2 2'3 l ^ /v n 7 2*3 
and consequently 

(1 + I)"<2 + I+ /- + ... + -— , 

v n> 2 2*3 2*3.. .n 

and with greater reason 

( 1 + ^) n<2 + ! + -i + i + --- + 2^- 

Now 2 + 2* + • " • + 2^ = 2" i 1 + 2 + 2^' ' ' + 2^)> 
and (Treat. § 63. ex) 

11 1 1 

1 + — 4. _ 4. . . . j _ = 2 — < 2 ; 

therefore Ave have at once, 

(1 + I)n >2 ,and(l+ -)»<3; 

and although by increasing n, (l -\ \ n increases also, still it 

cannot increase so much as to become = 3. The value, there- 
fore, of (l + — ) n is represented by a number between 2 and 3. 

The letter e is used to represent this number, and its approxi- 
mate value obtained by substituting in the above formulas, 
for n, positive numbers greater and greater, is 2,7182818 ; 
that is, 

e = 2,7182818 . . . 

Hence, however great the value of n may be, even if it be 
4 



48 PRINCIPLES OF INFINITESIMAL CALCULUS. 

infinite (infinity is represented by the sign oo), in which case 

— is infinitesimal, (1+ — ) n will be equal to e; nay, then only 

does it acquire the exact value of e when n = oo. Thus we 
have 



v 00 ' 



or, representing the infinitesimal by w, 

(1 + w )« = e . 
1st. Therefore, The binomial (1 + w), m which w zs infinitesimal, 
raised to the infinite power — , gives for result e = 2,7182818 . . . 

Take now with (1 — w) w , an infinitesimal quantity a, 

such that 1 — « = , and consequently — w = — ; 

l + a 1 + a 

we shall have 

JL 1 l+a -J l + a 

(i-« ) - = (r+^) a =—i— !+_«=(! + «) - 

l + a _1 , «_ 2 

but (1 + a) « = (1 + a)« « = (1 + a) (1 -f a)T, 

l j_ 

and (1 + a) « = e, therefore (1 — w) « = 6 + ae, and neglect- 
ing the infinitesimal, 

(1 — w) w = e. 

2d. That is, Tlie binomial (1 — w), in which u is infinitesimal, 

raised to the infinite negative 'power , gives for result 

CO 

e=2,7182818 . . . 

We admit the circle to be the limit of an inscribed polygon 
the number of whose sides increases indefinitely ; we must con- 



DIFFERENTIAL CALCULUS. 49 

sequently admit also the circle to coincide with the inscribed, 
polygon of an infinite number of sides. But the sides of this 
polygon are necessarily infinitesimal. Therefore an infinitesi- 
mal chord in the circle (the same may be said of any other 
curve) coincides with the arc. Let us now represent by 2/3 
the infinitesimal arc coinciding with this chord. The chord 
being equal to 2 sin (3 } we shall have 2 sin (3 = 2 j3 f and con- 
sequently — — = 1. 

3d. That is, The ratio between an infinitesimal arc and its 
sine is equal to 1. 

V. Differentials of algebraic functions. 
Let/(z) represent any of the following functions : 

I. a db z. II. az, in. — , IV. z. a 

z 

According to the definition (in.) we shall have 

d (a ± z) = a rb z ± dz — (a db z) = =b dz ; 
that is, 

I. The differential of a ± z is the same as that of ± z; and 
since supposing z = <p (#), we infer d (a ± <p (x) ) = ± d <p (x) y 

The differential of a db cp(x) is the same as that of <p (x). 
From the second we have 

daz = az -f- adz — az = adz ; i. e., 

II. The differential of the product of a variable by a constant 
is the product of the constant by the differential of the variable ; 
and since making z = <p (x) we obtain 

da cp (x) = adcp (#). 

So also the differential of the product of a function of a by a 
constant, equals the product of the same constant by the differen- 
tial of the function. 

We have from the third 

a a a _ adz > . 

z z + dz Z JT 



50 PRINCIPLES OF INFINITESIMAL CALCULUS. 

III. The differential of a constant divided by a variable equals 
the negative product of the constant by the differential of the 
variable divided by the square of the same variable ; and taking 

The differential of a constant divided by a function of x equals 
the negative product of the same constant by the differential of 
the function y divided by the square of the function. 

Lastly, from z a we obtain, in the supposition that a is a whole 
number, 

dz a = (z + dz) a — z a = z a + az* 1 - 1 dz -f — I~ \ z"- 2 dz~ 2 

~y . . . ~~~ Z * 

Or, neglecting the terms multiplied by the differentials of the 
orders superior to dz, 

dz a = az a ~ 1 dz; 

from which, regarding z as a simple independent variable or 
as a function of x, we infer that, 

IV. The differential of the power of a variable x or of the power 
of a function of x equals the product of the index by the given 
poioer diminished by one unit, and all multiplied by the differential 
of the variable or of the function. The same rule is applicable 
to the case of a being any number whatever. From the first 
equation, dz a = (z + dz) % — z a , we deduce the following: 

dz- =°*(l + -f — z«'=z* [(1 + -) a - 1], 

dz 
Now (l + — ) a — 1 is an infinitesimal which can be repre- 

v z ' 

sented by 6, and thus 

Applying logarithms to this last equation, we have 
al(l + ^) =1(1 + 6) 



DIFFERENTIAL CALCULUS. 51 



1(1 + -) 

and a /7/1 , .. = 1. 

(1(1 + 6) 

But from the first equation we have also 

1(1 + 6) 
from which 

l(l + $) ll (!<+*) 

dz? Z* , v z ' dz v z ' 



Now ± I (1 + j) XV(I + ^)* and 1/(1 + 0) = Z(1 1 ~°> 
and cfe being infinitesimal, and consequently — infinite as well 

CLZ 

as — , it follows from the 1st theorem of the preceding number 
that 

(l + 2?)*=(l + fl)e =e; 

efe' 1 
hence —,- = a^ —1 , 

and consequently 

c?z a = az a ~~ 1 dz, 

whatever be the numerical value of the constant a ; therefore 
The differential, etc. 

The functions whose differentials we have found embrace 
all the cases of algebraic functions. Let us now find the 



52 PRINCIPLES OF INFINITESIMAL CALCULUS. 

VI. Differentials of transcendental functions. 

Let/ (2;) represent any of the following functions: 

1. Iz, 11. a% ill. sin z, iv. cos z, v. tg z y vi. cot z y 
Vii. arc (sin = z), viii. arc (cos == z) y ix. arc (tg — z) y 
' X. arc (cot = z). 

Arc (sin = z) signifies an arc whose sine is z, and in like 
manner the last three functions signify an arc whose cosine or 
tangent or cotangent is z. 

Taking now the differentials, we have from the first 

dlz = I (z + dz) — Iz ; but I (z + dz) — Iz = I ( ] ; hence 

dlz = I (1 + -), 
j dlz 1 7 /,„ , dzs 

and & = s tl + 7 ; 

Z 

hence dfe = -dz, 

2! 

and taking e for the base of the logarithms, as is commonly 

done, 

, 7 dz 
dlz — — ; 

z 

hence regarding, as usual, z as a simple variable or as a func- 
tion of another variable x y 

I. The differential of the logarithm of a variable or function 
of a variable is obtained by dividing its differential by the same 
variable or function. 

From a? we have 

da z = a z + d * — a 9 — a z (a dz — 1). 
Now a dz — 1 is infinitesimal, and may be represented by 6, 
so that a dz = 1 + 6; and taking the logarithms 



DIFFERENTIAL CALCULUS. 53 

la 

hence da z = a z ' 6 = a z ' d — - r dz; 

1(1 + 6) 

W A U l ^ l ^ -la- 

^ 1 + ' } jl(l + *) 1(1+6)1 

therefore 

da? = a*Za dz ; i. e., 

II. The differential of the exponential quantity a z is the product 
of the same quantity by the logarithm of the root a multiplied by 
the differential of the exponent, whether it be an independent 
variable or function of another variable. 

We may come to the same conclusion by a more speedy 
process. 

Make a* — y, apply logarithms and take the differential, 
we shall have 

d lose a* = d lose y = ~ = — ; 

hence 

da z = a z d log a z = a z d • z log a = a z la dz. 

From the third and fourth functions we have 
d sin z = sin (z + dz) — sin z, 
d cos z = cos (z -f- dz) — cos z. 
And (Trig., p. 252) 

sin (z + dz) — sin z — 2 cos J(2z + dz) sin Jc?z, 
cos (z -f ds) — cos 2 = — 2 sin J(2z + cfe) sin \dz. 

But from theorem 3d (iv) we have sin \dz = \dz, and 2s + dz 
may be regarded as equal to 2z ; hence 

d sin 2 = cos 2 dz, d cos 2 = — sin z dz ; i. e., 

III. TAe differential of the sine is equal to the product of the 
cosine by the differential of the arc. 

IV. The differential of the cosine is equal to the negative pro- 
duct of the sine by the differential of the arc. 



54 PRINCIPLES OF INFINITESIMAL CALCULUS. 

From the fifth and sixth functions we have 

, , , sin z sin (z + dz) sin z 

d tg z — a = — -; — ■ — 5-: 

cos z cos (z + dz) cos z 

7 . 7 cos z cos [2 + cfe) COS z 

d cot 2 = d - — = - — 7 — r—ri — > 

sin z sin (2 + dz) sin 2 

hence (Trig., p. 253) 

, __ sin (z + cfe) cos 2 — cos (2 + efe) sin 2 __ sin (z-\- dz — 2) 

cos (2 + efe) cos 2 cos (2 + dz) cos 2 

sin c?2 



cos (2 + efe) cos 2 
But sin c?2 = dzy 2 + c?2 is equivalent to 2 ; hence 

dmz = — k-. 
cost 2 

By a like process we obtain 

j . dz 

d cot 2 = r-^- ; 1. e., 

sin 2 

V. The differential of the tangent equals the differential of the 
arc divided by the square of the cosine. 

VI. The differential of the cotangent equals the negative dif- 
ferential of the arc divided by the square of the sine. 

Calling y the arc whose sine or cosine is 2. with the equa- 
tions y = arc (sin == z), y = arc (cos = 2), we shall have the 
two following : 

2 = sin y, z = cos y, 

and dz — cos y . d,y, dz = — sin y ,dy ; 

hence dy or 

eZ arc (sin = 2) == — — > c£arc (cos = 2) = : — . 

cos y sin y 



2 „. 



Now cos y = v/l — sin 2 2/ = v'l — 2 2 , sin y — ^1 — cos 2 y 

VET?, 
therefore 

t / • \ dz . . s e?2 
a * arc (sin = 2) = — ,a'arc (cos = 2) = ; 1. e.. 



DIFFERENTIAL CALCULUS. 55 

VII. The differential of the arc whose sine is the variable z, is 
equal to the differential of the variable divided by the square root 
of(l-z'), 

Viii. The differential of the arc whose cosine is the variable 
z, is equal to the negative differential of the variable divided by 
the square root of (1 — z 2 ). 

Calling now y the arc whose tangent or cotangent is z, with 
the equations y = arc (tg = z), y — arc (cot = z), we shall have 
the two following: 

z = tgy, z = cot y, 
and (v., VI.) 

& _ d v_ dz d y . 

tW „ j iX4i — — = j 

cos y sin y 

hence dy or 

■j /■ , \ 9 f jl 7 az az 

a * arc (tg = z) = cos" 5 y dz — — „-- dz — - — ; ^— = 9 , 

v & ' * sec 2 y 1 + tg 2 y 1 + z 2 

1 dz 

c? arc (cot — z) = — sin 2 y * dz = — dz = 



cosec y . 1 H- cot i/ 

c?z 



1 + 



*2> 



i. e., 



IX. T/ie differential of the arc whose tangent is the variable 
z, is equal to the differential of the variable divided by (1 + 2 2 ). 

x. The differential of the arc whose cotangent is the variable 
z, is equal to the negative differential of the variable divided by 

(i + **>. 

VII. Differentials of the sum, product, and quotient of different 
functions of the same variable x. 

Let now the following functions of the* same variable x be 
given, 

u = F (x), y = f(x), z = <p (x), 

and let s be their sum ; i. e., 

s=F(x)+f(x) + <p(x), 
we shall have 



56 PRINCIPLES OF INFINITESIMAL CALCULUS. 

du = F (x + dx) — F (x), dy = f(x + dx) — /(a?), 
dz = 9 (x + cfo) — 9 (#), 
and also 

cfc = F (a; + dx) — F (#) +/(a? + dx) —f(x) -f 9 (a; + cfc) ■— 

9 (a?)J 
Therefore 

c?s = c?u + dy + c?2 ; i. e., 

I. The differential of the algebraic sum of different functions 
of the same variable is equal to the sum of the differentials of 
each function. 

Let now_p be the product of y by z ; i. e., let 
P = yz=f(x)x 9(a>), 
and consequently, 

now 

lp 2 =ltf+lz>; 

also (VI. 1.) 

dif = d £, dif = J 2 , dw =^, 

and (V. IV.) 

dp 2 = 2p dp, dy 2 = 2y dy, dz 2 = 2zdz; 

,, „ 2p dp 2y dy , 2zdz 
therefore 3- -^ = 1*2 ^ _, 

p 2 2/ 2 2T 

c?p cfy efe 
or -4f = 4 + — ; 

that is, since p = y z, 

d(yz) = zdy -f ^dz, 
or c? [/(a) 9 (a-)] = 9 (x) of (x) + ${$) d 9 (a?) ; i. e., 

II. TAe differential of the product of the two functions of the 
same variable is obtained by multiplying each function by the 
differential of the other and adding together the two products. 

It is known from algebra that a positive quantity raised to 



DIFFERENTIAL CALCULUS. 57 

a power indicated by any exponent, either positive or negative, 
gives always a positive result. Now the base of logarithms if 
positive in every system ; hence negative quantities admit < 
no logarithms but imaginary ones. To avoid the incon- 
venience of these imaginary logarithms, the equation p = y z 
has been squared in the process of the preceding theorem. 
To give an example of the same theorem, let y=x 3 and z = 
sin x. We shall have 

p = x s ' sin x and dp = d(x z ' sin x) = 3x 2 sin x dx + x 3 cos x dx. 
Let, lastly, q be the quotient — = -)~4' From q == — we infer 

Z (p \X) z 

y = q' z and 

dy = zdq + qdz ; 
hence 

zdq = dy — qdz = dy — — dz = — - — , 

and therefore do = — - — ~ — , 

zr 

jf( x ) _ 9 (fflff (?) tt/WI <P_fo) . • . 

V(^)"" [9W] 2 

in. Jfte differential of the quotient of two functions of the same 
variable is obtained by talcing the difference between the product 
of the denominator by the differential of the numerator, and the 
product of the numerator by the differential of the denominator , 
and dividing this difference by the square of the denominator. 

Let, for example, the quotient be , — V-%> we shall have 

log (x) 

, 2x* 6x 2 log (x) dx — 2x 2 dx 2x 2 (3 log (x) — 1) dx 

log (x) log 2 (x) log 2 X 

"VIII. Successive differentials and their orders. 

Let, for instance, y = x n y 

we shall have (V. IV.) 

dy = nx n ~ * dx. 



58 PRINCIPLES OF INFINITESIMAL CALCULUS. 

The differential dx of the independent variable being taken 
as a constant and always the same in the succeeding differen- 
tials ; from the above differential (which is another function of 
x) again differentiated, we shall have (V. H. IV.) 

d (dy) = n(n — 1) x n ~ 2 dx 2 . 

d (dy) is represented by d 2 y and the following differentials 
by d 3 y, d 4 y, etc. Thus following the same process, the first 
and the succeeding differentials of y = x n are given as follows : 

dy = dx n = nx 11-1 dx, 
d 2 y = d 2 x n = n(n — 1) x n ~ 2 dx , 
I. ]d 3 y=d 3 x n = n(n — l)(n — 2)x n - 3 dx 3 , 



d n y = d n x n = n (n — 1) (n — 2) . . . (n — (n — 1)) dx n - 

These successive differentials are calied also differentials of 
various orders, 1st, 2d, 3d, . . . nth. We may remark that in 
the above example the last differential is constant, and conse- 
quently d n+1 y = 0. 

The line placed above dx in the differentials of the second 
and following orders is to distinguish the power of dx from 

the differential of x raised to a power. Thus, whereas dx a , for 
example, represents dx raised to the power a, dx a signifies the 
differential of x raised to the power a. 

Let, secondly, y = a x , 

we shall have (VI. n.) 

dy = a x ladx, 

and consequently, ladx being constant, 

d 2 y = a x l 2 adx , etc., . . . i. e., 
fdy = da x = a x la dx, 
d 2 y = d 2 a x = a x l 2 a dx 2 , 



II. 1 



d"y = d n a x = a x l n a dx n . 
Let, thirdly, y = sin x, we shall obtain (VI. III. IV.) 



DIFFERENTIAL CALCULUS. 

dy = d sin x = cos x dx, 



59 



d 2 y == c? 2 sin re = — sin as dx , 
' c% == d 3 sin a; == — cos a? cfo? 2 , 
cZ 4 ^/ = d* sin a; = sin x dx , . . . 

Let, lastly, y — log (x). We shall have (YI/l.) 



7 dx 1 7 



hence (V. in. IV.) 



2 — 



7 , 1 -t~2 t 7 o dx 7— 2arcfa? 

ay = 5- a^ , and en/ = — rr dx = — r— = — 5-aar ; 

17 or ' -< or a? 4 a; 3 



hence 



"c?y = cMog(rr) = — , 
a? 



efo 



IV. ^ 



^ = d 2 • log (a;) = — -x, 

a? 

1 * 2 ~dx 

d 3 y = dnog(x) = — - 3 — , 

dfy = d 4 log (a>) = ^-— , 



d w y = c?» log (a?) = 



1-2-3... (71 --l)cfe w 



a?' 



In the last of which formulas the negative sign occurs when n 
is an even number. 



IX. Derivative functions and their orders. 

Let us resume the formula y = x n and its successive differ- 
entials (VIII. I.) We shall easily deduce from them the fol- 
lowing equations : 

dv d"y 

JL =z nx 71 " 1 , =~ = n(n — 1) x n ~ 2 , 



dx 



dx 



^M = n(n—1) (n—2)x n - 3 . . . ^ = n(n— l)...(n — (n— 1)), 



60 PKINCIPLES OF INFINITESIMAL CALCULUS. 

all of which, the last excepted, are functions of x, and being 
derived from the original function y = x n , they are called 
derivatives of that function. 

Let now y = f(x) be any function of x. The first, second, 
and following derivatives of this function are represented by 
f(x),f"(x), .../«(*); i.e., 

ax ax ax 

and are called derivatives of the 1st, of the 2d, ... of the nth 
order. 

Thus, The derivative function of any order is given by the 
ratio between the corresponding differential of the primitive func- 
tion and the corresponding power of the independent variable. 

But from the last equations we obtain 

dy — f (x) dx, d 2 y — f n (x) dx" y . . . d"y = /(") (x) dx n ; 
hence 

The differential of any order of a given function is given by 
the product of the derivative of the same order by the correspond- 
ing power of the differential of the independent variable. 

Since in these last equations/' (x),f /f (x), ... or the equiva- 
lent ratios -¥-, =J- 2 , . . . perform the office of coefficients ; they 
dx dx" 

are also called differential coefficients of various orders. It is 
plain, from what precedes, that the derivative functions are 
obtained by finding the successive differentials and omitting 
in them the differential dx of the independent variable and its 
powers. 

X. Maclaurin 's formula. 

Suppose the function f(x) to be capable of being developed 
into a series arranged according to the increasing powers of x 
as follows : 

f(x) = A + A x x + A 2 x 2 + A 3 x s + A 4 x* -f ... 

in which A and the coefficients A 1? A 2 , . . . , which are inde- 



DIFFERENTIAL CALCULUS. 61 

pendent of x and constant, are unknown. The object of Mac- 
laurin's theorem or formula, as it is called, is to find these 
coefficients. 

It follows from the definition of the differential (III.) that 
the differential of a constant quantity is equal to ; 

hence <^A = dA x = dA 2 = . . . = 0, 

and d2A 2 = d2 • 3A 3 = . . . = 0; 

hence, from what has been said in the preceding number and 
from the rules II. and IV. of No. V., we shall have with the 
primitive function 

fix) = A + Ai x -f A 2 x 2 + . . . 
the derivatives 

/' (x) = A x + 2A 2 z + 3A 3 £ 2 + 4A 4 r* + . . . 

/" (x) = 2A 2 + 2 • SA s x + 3 • 4A 4 ar 2 + . . . 

/"' (a?) = 2 • 3A 3 + 2 ■ 3 ■ 4A,x + . . . etc. 

In all these equations x may have any value, without affect- 
ing the constants A , A 1? . . . , but making x = in the primi- 
tive and derivative functions we have 

/(0) = A* /' (0) = A„ f" (0) =2A 2 , /{" (0) = 2 • 3A 3 , . . . 

hence the constants A , A 1? . . . are given by the primitive 
function and by the derivatives, making in them x = 0. Sub- 
stituting the values of the constants thus obtained in the primi- 
tive, we shall have 

/ (*) =/(<?) + s/! (0) + jf" (0) + ~f" (0) + . . . 

which is Stirling's formula, more commonly known as Mac- 
iaurin's. It answers the purpose of developing functions 
into series according to the increasing powers of the variable. 
Let us see some applications, and let, first, f{x) = e x . We 
then have (VIII. II.) /' (x) = e x , and consequently also/" (x) 
— f" f ( x ) — • • • = gX > an( i therefore, taking x = 0, / (0) =■« 
/'(0)=/"(0) = ...= l; hence 



62 PRINCIPLES OF INFINITESIMAL CALCULUS. 



,2 £ „4 



I.) «-.,l + * + _ + _ + _ 4 + ... 

and making in this formula x = 1, 

e=2 + l + 2 1 3 + 2^4 + "- 

by means of which the value of e (IV.) can easily be obtained 

as nearly as desirable by increasing the number of terms of 

the series. 

Let, secondly,/ (x) = sin x. We have (VIII. ill.) 
f (x) = cos x, j" (x) = — sin x } j' n (x) — — cos x, 
j" ff {x) = sin x, . . . 

and making x = 0, 

/(0) = 0, /'(0) = 1, /"(0)-0, /'"(0) = -l, 
/""(0) = 0, /M(0) = 1... 
Therefore 

(II.) sina = z-^ 3+ ^--... 

Let, thirdly, / (x) = cos x y we shall have 
/' (x) = — sin x, f (x) = — cos x, f ,n (x) = sin x, 
f(*y) (x) = cos x y ;» . . 
hence 

/(0) = 1, /'(0) = 0, /"(0)--l, /'"(0) = 0, 
/« (0) = 1, . . . 

and consequently 

X 2 X 4 

(in.) cos x = 1 — *- + 2T3T4 — • • • 

XL Taylor' s formula. 

Let us take the function f(x -f A), in which A represents an 
addition made to x, either positive or negative. Considering 
this addition as variable and the undetermined quantities, B , 
B 1? B 2 , B 3 , . . . independent of h in the supposition that 
f(x-{-h) is capable of being developed into a series arranged 



DIFFERENTIAL CALCULUS. 63 

according to the increasing powers of h, we may, as in the pre- 
ceding paragraph, represent / (x + h) by the series B + B L /i 
+ B 2 /t 2 + B 3 /i 3 + . . . , in which B , B 1? B 2 , ... necessarily 
depend on x. Supposing now h variable, and taking the suc- 
cessive derivatives of f(x 4- h) relatively to h alone, making 
then in the primitive and in the derivatives h = 0, we shall 
find, as for Maclaurin's formula, the equations 

. B =/(*), B, =/' (x), B 2 =1/" (x), B 3 = l^f" (x), . . . 
and consequently 
/(» + *) -/(*) + hf (x) + I /" (x) + ^Lf" (x) + ... 

which is Taylor's formula, by means of which we obtain the 
second state of a function developed into a series of terms, 
arranged according to the increasing powers of the addition h 
made to the variable in the function / (x). 

1st. Let, for instance, / (x) — ^x, we shall have (V. IV.) 
1 l 1 1 3 1 



2 2^x 4 4^/ x v 

hence, if in the given function vx we change x into x -f h, 
Taylor's formula will give us 

v x -j- h = v x -f ■ — j=. j= + 



2^x 8^x s lQ^x 5 

2d. Let also f(x) = — , we shall have (V. 111. and iv.) 

v x 



2*^0? Wx 5 2'^x 7> 

3-5-7 



/M (x) = 



2 • 8*V 



64 PRINCIPLES OP INFINITESIMAL CALCULUS. 

Therefore, according to Taylor's formula, 

1 1 1 h 3 tf 3-5 h 3 



3-5-7 A 4 

2 • 4 • 6 • 8 v/^9 

making now # = 1, and /i = — z 2 , 



•1=? '2 '2-4 2-4-6 

3-5-7 



2-4-6.8 



2?+... 



as we would obtain by applying to — « or to its equal 

1 
(1 — z) 2 , the development of the Newtonian formula. 

Taylor's formula rests on the fact that /(a; -f- h) is capable of 
being developed into a convergent series having f(x + /i)for 
limit of its convergency. If this fact be not verified, the 
formula must necessarily fail to give the value of the function 
represented by an unlimited series. This is precisely the case 
in the last example if z be supposed greater than 1. For 

whereas — , for any finite value of z has a finite and 

v 1 — z 2 

fixed value, the series increases in value in proportion as it 
increases in the number of terms. But if z <^ 1, then (Alg. 
§ 47) the series 1 + z 2 + z* -f z 6 -f • • • indefinitely pro- 
tracted, has for its value r: it is, therefore, a conver- 

1 — zr 

gent series. With greater reason, therefore, the second mem- 
ber of the last equation is a convergent series, whose terms 
are the same as 1 + z 2 + z 4 + . . . , all multiplied, except 
the first, by a constantly diminishing fraction. For the coeffi- 
cient of the third term is the coefficient of the second multi- 
plied by a fraction, the coefficient of the fourth term is the 
coefficient of the third multiplied by another fraction, etc. 



DIFFERENTIAL CALCULUS. 



65 



Thus, in the supposition of z < 1, the function 



vT 



is ex- 



*l 



aetly represented by the series of the second member of the 
above equation indefinitely protracted. 

XII. Maxima and minima of functions of a single variable. 

Any real function f{x) can always be represented by the 
ordinates of a curve CC r (Fig. 18) corresponding to abscissas 
representing the different values of x. Let now h be a posi- 
tive and infinitesimal quantity, and let the value x m of x be 
represented by the abscissa AK. From K take KK' == KK" 
= A. If we find that the ordinate KM —f(x m ) is greater 
than the preceding K'M' and the following K/'M 77 , it is called 
a maximum of f(x); if, on the contrary, KM be found less 
than K'M' and K^M", it is a minimum of the same function ; 
i. e., any function f(x) will be a maximum or a minimum for 
a particular value x m of x according as we shall have 

f{Zm)>f(%m±h), 

or f{x m ) <f(x m -±zh), 

or in other terms, according as we shall have 

Cf(x m ±:h)-f{x m )<0, 
(I.) \ov 

(f(x m ±h)—f(x m )>0. 
Taylor's formula enables us to find whether and when these 
conditions are verified ; for in the case of h being positive we 
have from this formula 

[f{x m + h) -f(x m ) = hf> (x m ) + J/" (O + 



¥ 



J'"{x m )+ 



2-3' / x mj 2-3-4 
(n.) \ and when h is negative 



J \%m) i • • • 



K- 



f(x m — h) —f(x m ) = — hf (x m ) + - f' (x m ) 
— oTq/ \ Xm ) "+" o~."o"."7/ 1T \ Xm J * * ' 



2-3' 



2-3-4' 



66 PRINCIPLES OF INFINITESIMAL CALCULUS. 

Now the sum of infinitesimal quantities of different orders can 
have no other sign but that by which the infinitesimal of the 
lowest order is affected ; hence the first members of the equa- 
tions (n.) cannot have the same sign unless j' (x m ) = ; but 
if j" (x ol ) does not vanish withy 7 (x m ) 7 then the first members 
of the two equations will be affected with the same sign, posi- 
tive or negative, according as / /; (x m ) > or < 0, saidf(x m ) 
will be a minimum in the first case, and a maximum in the 
second. In case that with/' (x m ) — 0,/" (x m ) also should be 
= ; then, in order that / (.r TO ) be a maximum or a minimum, 
j'" (#m) also must vanish ; and supposing that / IV (x m ) does 
not vanish with the preceding derivative, f(x m ) will be a 
maximum when/ IV (x m ) <[ 0, and a minimum when/ 17 (x m ) > 
0, etc. In general, let f^ (x m ) be the first derivative which 
does not vanish. If n be an odd number, f(x m ) is neither a 
maximum nor a minimum. If n be an even number, then 
/ (x m ) is a maximum when /(") (x m ) <C 0, and a minimum when 
/ (n) (x m ) > 0. But the same value x m which, in this case, 
makes a maximum or a minimum of the primitive function 
f(x, n ), fulfils the equation /' (x m ) =/" (x m ) = . . . =f»~V (x m ) 
= 0, therefore, taking only the first and last member. The 
values z m of x which can render fix) a maximum or a mini- 
mum must be looked for among the roots of the equation 

which equation must always be verified whatever be the index 
(n) of the derivatives of a higher order than the first, which 
does not vanish. 
Let, for instance, 

f{x) = x 3 — 6x 2 + 9x — 3. 

To see if this function admits of a maximum or a minimum 
let us make the first derivative equal to zero ; i. e., /' (x) = 
3x 2 — 12ic -f 9 = 0. If there is any value capable of making 
a maximum or a minimum off(x) it must certainly be among 



DIFFERENTIAL CALCULUS. 67 

the roots of this equation, which is reducible to the following : 

x 2 — 4x + 3 = 0, 
and which resolved gives us 

x m — 1, and x m = 3. 

Now the second derivative is, in our case, f ,f (x) = 6x — 12, 
which does not vanish by substituting in it for x t x m = 1, x m 
= 3. It is, besides, negative for the first of these two values, 
and positive for the second; therefore the function of x 

x 3 — 6x 2 + 9a; — 3 
acquires a maximum value when x = 1, and a minimum when 
x = 3. With the first of these values substituted we have 
f(%m) = 1, with the second f(x m ) = — 3. 

XIII. Values of functions which assume an undetermined form. 

fix) 

The ratio F (x) = -~-{ may assume the undetermined forni 

j <P (*) 
— , when for a particular value of x both functions / (x) and 

cp (x) become zero. We may ask if such a form can corre- 
spond to a definite value, and how this value can be known. 
The finding of this value will be a reply to both questions. 

From the given equation we have F (x) cp (x) — f(x) = ; 
hence (VII. I. n.) the derivative 

' <p (x) ¥' (x) + F (x) <f'(x)—f (*) = 0. • > 

But there is, by supposition, some value of x which makes 
cp (x) = 0. Substituting in the above derivative this particu- 
lar value of x, we shall have, in this case, F (x) cp f (x) = f (x), 
and consequently for the same value 

' V ; V cp ( X ) / <p' {x) 

But if the derivative /' (x), cp f (x) should also become equal to 
zero for the same value of x, and so likewise the following 
derivatives till the nth order exclusively, we would then have 



68 PRINCIPLES OF INFINITESIMAL CALCULUS. 

K } ~ 9 (x) ~ " <pN (a,) ' 1b e *> 
I. TAe tfrue vafoie q/" the ratio between two functions which 

assume the undetermined form — when a particular value of x 

is placed to them, is given by the ratio between the derivatives of 
the same order which are the first of those which do not vanish 
simultaneously when the same particular value of x is placed in 
them. 

Let, for example, / (x) = 1 — cos x } 9 (a?) = a? 2 , we shall 
have 
f {x) = sin x, (p f (x) = 2x, f" (x) = cos x, 9" (x) = 2, 

and consequently, first, F (x) = — -, which becomes -- 

gin rn C\ 

when x = 0. Secondly, F (x) = - , which also becomes — 

Ax U 

cos X 
when x = 0. Lastly, F (x) = — — = — when x = ; hence 

A A 

when x = 

1 — cos x 1 



F(4 = 



ar 2 2 



The ratio F (x) = -y4 may also assume the undetermined form 
_o? for some particular value of x. In this case we shall have 
= 0, and also -f-r~\ : ~r\ = ~K * Now the deri- 



vatives of -—r and — — are (V. m.) — - VH and rH; 

f(x) cp(x) /-(a?) 9 2 W 

therefore, from the preceding theorem, 

/M>(*)\ " 0^ f (xj ' ? 2 (*)' 

f 2 fa) 

and multiplying each member of the equation by -yy-y 



DIFFERENTIAL CALCULUS. 69 

Therefore, if for the same value of x the derivatives of f(x) 
and 9 (a;) should become infinite till the nth. order exclusively, 
the ratio of the given functions will be determined as in the 
preceding case by the formula 

W v(xj~ *>~ <?<■•» {x)' ' 

II. The value of the ratio of two functions which for a par- 
ticular value of x becomes — is given by the ratio of the two first 
derivatives of the same order which neither vanish together nor 
together become infinite with the same value of x. 

Let, for example, / (x) = log (x) and <p (x) = cot (x). Sup- 
posing x = 0, we have log (x) = — 00, cot x = 00 ; thus for 

the particular value of x. J —~ == - ; = — ^. Taking 
F ' <p (a?) cot (a?) °° 5 

now the derivatives, we obtain (VI. 1. VI.) /' (x) = — and 

x 

f (x) sin 2 x 

q>' (x) = r-s— ; hence -~-( = equal to — .for a? = 

v ' snrar $' (x) x * 

; but the derivative of — sin 2 x — — 2 sin a; cos x and the 

-, . , . n • -, i /' (x) 2 sin x cos a? 

derivative oi x is 1 : hence ''-H = — = = 0. or 

<p'(«) 1 V 

F (a;) = - g -^ = - ^ = when a = 0. 

cot x °° 

It may also happen that one of the functions becomes for 

a certain value of x, and the other becomes go for the same 

value. It is easy to see what, in this case, would be the deter- 

f (x) 
mined value of J —^!, but the value of the product F (x) = 

y(x) r K 3 

f(x)cp(x) would have the undetermined form ' go. Now 
/(*) 9 (*) =/(*) :± } = l = 9 (x) :/ y = » ■ Thus the 



70 PRINCIPLES OF INFINITESIMAL CALCULUS. 

present case is reducible to one or other of the two preceding, 
and 

in. The product of two functions which for a particular 
value of x becomes * oo, is obtained from the ratio between the 
derivative of one of the factors .and the quotient resulting from 
unity divided by the other factor. 

Thus, let f (x) = x and <p (x) — log (x). The product 
x log (x) becomes — * oo when x = 0. Now the derivative of 

log (x) is — and the derivative of — is - o9 therefore when 

x = 

f(x)'q> (x) = x log x = — * co = — : = — x = — 0. 

Besides the preceding, three more undetermined expressions 
deserve our attention, i. e., 0°, oo°, 1°°, which the function 

FW = [/W]'H 

assumes, when, for certain values of x both functions / (x) and 
<p (x) become = 0, orf(x) = go and <p (x) = 0, or, finally f(x) 
= 1 and (p (x) = co. It might also happen that with f(x) = 
we would have o(x) == 00, from which would result another 
undetermined expression 0°°. But let it be observed that 
from the given formula we deduce I F (x) = 9 (x) I [/(#)] = 

WMl hence . *&Lp® 

(p (x) F (x) = e (p(x) ; for from this as well 

as from the given formula we deduce I F (x) = <p (x) I [/(#)]. 

log [/(*)] 



Thus F (x) = [/ (a?)]* (*) = e <p (a). 

Commencing with the last case, w T e have 

F (&•) = 0°° = e ° = in every supposition. 

In the first case F(a>) = 0° = i~~" for IPlIZ^)] « — ig, 

<p(x) 







<P 0) 











u 


a 


iC 




(( 

0" 



DIFFERENTIAL CALCULUS. 71 

In the second, ¥{x) = oo° = 6 " for fetX^ = + «>. 

In the third, F(aO=l°° = e 

Therefore the determination of the expressions 0°, oo°, l 00 is 
obtained by the same expedient by which the preceding unde- 
termined expressions have been determined. To give an 
example of each of the last three cases, let, 
1st, f(x) = 9 (x) = x. We shall have 

log (afl 

F (a) = <c* = c — = 0° = e~ » when a? = 0. 
a; 

But — is the derivative of Iff a?, and h the derivative of — : 

O/ «|/ «/ 

, loff'(ar) oo 1 1 ^i 

hence - ° N = — £J = : -^ = — a? = — when t 

1 °° a; x 1 

x 
= and 6 _0 = -^ = 1 ; i. e., when x = 
F (a?) = a? = 0° = 1. 

2d. Let / (x) = x, cp (x) = — , and suppose x = oo, we shall 

L_L l0giC » I 

have F (a?) = x x = oo° = e x = e 00 when x = oo. But — is 

a? 

the derivative of log (a?), and 1 the derivative of x ; hence 

logO*) = » !a l«o when a; = oo. But e° = 1. Therefore 
a; w . a? 

when x = oo 

F (a?) = #*"= oo° = 1. 

3d. Let f(x) = Xj <p (a?) = . Taking a; = 1 we obtain 

J. — - ~ x 



72 PRINCIPLES OF INFINITESIMAL CALCULUS. 

_1_ log Q) 0^ .. 

F (x) = x l ~ x = l 00 = e 1 ~ x = e°. Now — is the derivative 

a; 

of lg (#), and — 1 the derivative of 1 — x ; hence — — - = — 
== = — 1 when x = 1 ; i. e., when x = 1 

— 1 

e 

XIV. Cho7*d of an infinitesimal arc of a continual curve. 

Let DABD' (Fig. 19) be any continued curve. Divide the 
arc into three parts at pleasure, DA, AB, BD, and draw 
the corresponding chords, producing the first to T, and 
the last until it reaches the first in C. From this con- 
struction we have TCB = CAB + CBA. Calling s this 
angle, and designating by a, b, c the sides CB, CA, AB 
of the triangle CAB, we have (Trig. §§ 20 (e 4 ) and 12) 

c 2 = a 2 -f b 2 -f 2ab cos s. 
And since (Tr. § 20) cos s = 1 — 2 sin 2 J s also, 

c 2 = (a -f- b) 2 — 4ab sin 2 §s, 
easily reduced to the following : 

c 2 ., 4a& . 9 ., 

= 1 — v ,-t-t- sin z is. 



(a + bf (a + b) 2 2 ' 

in which the coefficient y - Th is equal to 1 — ( - y ) 2 ; i.e., 

(a+ b) 2 4 W& ; ' ? 

less than unity. 

Now, in a continual curve the smaller the arc DABD' 
becomes, the smaller also become the angles CAB, CBA, and 
when the arc becomes infinitesimal the angles also become 
infinitesimal, and s likewise, which is equal to their sum. 
Therefore, in this supposition, sin 2 Js is an infinitesimal of the 
second order, which, in the last equation, being besides multi- 
plied by a coefficient < 1, can be suppressed, giving us thereby 



DIFFERENTIAL CALCULUS. 73 

C 2 

— - = 1, or c = a 4- b. But if, when the arc DAmBD' 

(a + b) 2 

is infinitesimal, the chord AB does not differ from the sum of 
the sides AC, CB, it differs much less from the arc AmB sub- 
tended by it. Hence in any curve An infinitesimal continual 
arc and its chord coincide with each other, or their ratio is equal 
to unity. 

It follows from this theorem that The chord may be taken 
instead of the corresponding infinitesimal arc ; and since, in this 
case, the chord necessarily coincides with the tangent, it fol- 
lows also that Any curve may be regarded as a polygon of an 
infinite number of infinitesimal sides which produced will be as 
many tangents of the different points of the curve. 

XV. Tangent, subtangent, normal and subnormal of any plane 

curve. 

Let CC (Fig. 20) represent any plane curve referred to the 
orthogonal axes AX, AY, and let y = f{x) be its equation. 
Let also TT' be the indefinite tangent of any point M of the 
curve, and MP a perpendicular to the tangent from the point M 
of contact, the co-ordinates x and y of which are AK, KM. 
The segment MT of the indefinite or geometrical tangent of 
M, contained between the point of contact and the axis of 
abscissas, is called tangent of the point M. In like manner 
the segment MN of the perpendicular MP contained between 
the same point of contact and the axis of abscissas, is called 
normal of the point M. The tangent is represented by t y the 
normal by n. Of the two segments TK, KN of the axis of 
abscissas, measured from the points met by the tangent and by 
the normal to the ordinate of the point of contact, the first is 
called subtangent and the second subnormal of the point M, 
and are respectively represented by t, and n,. 

To determine the length of these four functions, observe, 
first, that calling X, Y the co-ordinates of the tangent referred 
to the axes of the curve, and X', Y' the co-ordinates of the 



74 PRINCIPLES OF INFINITESIMAL CALCULUS. 

normal, and (tx) the angle MTX, since both lines pass through 
the point M, and one is perpendicular to the other, we have 
(A. G. III. 2d, 3d) for the equations of these lines 

Y-y=(X-x)tg(tx), Y'-^=-(X'-*) * 

tg (tx) 

Take now from M the arc MM' infinitesimal, whose co- 
ordinates AK7, K'M' will be respectively represented by 
x + dx, y + dy, and drawing from M on M'K', MD parallel 
to AX, we shall have also MD = dx, DM' = dy. The arc 
MD, being infinitesimal, may be regarded as rectilinear and 
coincident with MT', therefore (IX.) 

tg (tx) = tg M'MD = p- = f(x); 

hence from the preceding equations 

Y-y=(X-x)f(x), Y'-y--(X'-»>^ 



/' (*) 

The abscissa X corresponding to Y = is — AT, and the 
abscissa X r corresponding to Y f = is AX ; hence, making 
in the last equations Y and Y r — 0, we shall obtain 

AT + *=^/_, m-x = yf{x)', 

i.e., t,=j/r^, n f =yf(x); 

and since from the right-angled triangles MKT, MKX 

MT = 4/iQf 2 + tS MX = Jffn-lf, 

so also, for the values of the tangent and of the normal, 
t = ^y 2 + t, 2 j n = ^y 2 -f n 2 , 

or t = y\/l+j^- )y n^ 9 >'!+/*($. 

These and the preceding formulas, being altogether general, 
can be applied to the lines of the second order. 

1st. Functions of the 'parabola. ' Commencing with the para- 
bola, whose equation is (A. G. V.) 



DIFFERENTIAL CALCULUS. 75 



y = ^2px, 
i. e., f(x) = \/2px, we shall have (V. IV. and IX.) /' (x) = 

— %■ — = H-- hence, since in the parabola (L. c.) p = x + \p 
\/2px y 

designates the distance of the focus from the point (x, y) of the 

curve, we shall have 

V 2 
t. = — = 2x, n, = p ; 

P 

i. e., the subtangent of any point of the parabola is equal to the 
double of the abscissa of that point, as we have already found 
(A. G. VIII.) with a different process, and the subnormal is 
constant and equal to the semiparameter. Concerning the tan- 
gent and the normal, we have from the preceding general 
equations 

t = y\/l + K 2 = ^2px + 4x 2 = </4x (x -f ^p) = V4#p, 
p 

n = y\f 1 +-^-= ^2px + p 2 = V*2p (x + Jp) = ^2pp ; 

i. e., The tangent of any point of the parabola is mean geometri- 
cal proportional between the focal distance of the point of con- 
tact and the quadruple of the abscissa of the same point. The 
normal of any point of the parabola is mean geometrical' pro- 
portional between the focal distance of the same point and the 
parameter. 

2d. Functions of the ellipse. The ellipse referred to its 
own axes is represented (A. G. XI.) by the following equa- 
tion : 

y = — ^d 2 — x 2 ; 
J a 

hence, in this case, /(a;) = — ^d 2 — x 2 and (V. IV.) f (x) = 

a 

bx 
— ^T/ ~"2 Y > anc * f rom tne e( l ua tion (2) (A. G. XI.) 



76 PRINCIPLES OF INFINITESIMAL CALCULUS. 

a 2 — b 2 = a 2 e?. From the above general formulas we obtain 
for the ellipse 

a 2 — x 2 _ a 2 b 2 

i. e., taking into account only the absolute value, The sub- 
tangent in the ellipse is equal to the difference between the ab- 
scissa of the point of contact and the square of the transverse 
semiaxis divided by the same abscissa. Hence the subtan- 
gent is independent of the conjugate axis. TJie subnormal is 
equal to the product of the abscissa of the point of contact by 
the square of the conjugate divided by the square of the trans- 
verse semiaxis. Consequently the ratio of the subnormal and 
the abscissa of the point of contact is constant in the ellipse. 

The general formulas of the tangent and normal become 
for the ellipse 

t = y v/l + gKtZgS = 1 v'&V + a 4 — a¥ 2 = 

J v 6V bx 

OX 

but a 2 — b 2 — a 2 e 2 , hence 

b x 

v / 'a 2 — eV: 



v^a 2 — x 2 



but y = — (vV — x 2 ) ; hence 



« 



b 



n— — */ a 2 — eV 
a 



i. e., The tangent and the normal of any point of the ellipse are 
given by formulas analogous to the equation of the curve, chang- 



DIFFERENTIAL CALCULUS. 77 

ing in the latter for both of them x 2 into e 2 x 2 , and for the tangent 

77 m • b . J a y 

also the coefficient — into -,— • — • 
M a b x 

3d. Functions of the hyperbola. The equation of the hyper- 
bola referred to its own axes is (A. G. (3) XIX.) 

b 2 

2/ 2 = - 2 (* 2 -a 2 ); 
Li 

hence /' (x) = , and observing that from the equa- 

a ^x 2 — a 2 

tion (2) (A. G. XI^L) a 2 + b 2 = a 2 e 2 , following the same pro- 
cess as for the ellipse, we shall find 



and 



x 2 — a 2 _ a 2 b 2 

t f — — x — * — , n/ — — 5 X) 

x x a 



t == -r- . — ^e 2 x 2 — a 2 , n = — *Se 2 x l — a 2 , 
ox a 



from which follow exactly the same inferences as for the 
ellipse ; hence the preceding conclusions w T ith regard to the 
functions of the ellipse are applicable to those of the hyperbola. 

XVI. Differential total and partial of a function of different 
'independent variables. 

So far, we have supposed functions depending on only one 
variable. Let us now pass to see how differentials of functions 
containing more than one independent variable can be ob- 
tained. Let, for example, jx be a function of the variables x, 
y, z 9 independent of each other, and having the following 
form : 

x log (y) 



f* = 



sin z 



The total differential of |x is the difference between the value 
of the given function and that which the same function assumes 
when we change all the variables by an infinitesimal quantity, 



78 PRINCIPLES OF INFINITESIMAL CALCULUS. 

or (3) the difference between the first and second state of the 
function when each one of the variables undergoes an infini- 
tesimal change. Thus, representing by dp the infinitesimal 
change of p, resulting from those of all the variables, we shall 
have 

, _ (x + dx) log (y + dy) x log (y) 
sin [z -f dz) sin z 

If all the variables are not changed, but only one of them, 
the change which p undergoes in consequence of that of the 
variable is evidently a partial differential. It is represented by 
d x p when x alone is changed into x 4- dx, or by d v p, d z p when 
y or z alone is changed. In this supposition, we shall have 

d x p = + dx ) lo g (y) _ g l og (y) d ■ g % (y + <%) _ 

sin z sin 2 ' w sin z 

x lQ g (y ) j- « l og (y) »log(y) 

sin z sin (z -f dz) sin 2 

More generally, If the function p be represented by / (x, y, 
2), the total and partial differentials will be expressed as fol- 
lows : 

dp =f(x + dx,y + dy, z -f dz) —f(x, y, z), 
d x p =f{x + dx, y, z) —f(x, y, z), d y p = f (x, y + dy, z) — 

f 0, y, *\ d& = / 0, y, * + <&) —f 0, y, «)• 

Concerning the partial differentials, they are obtained ex- 
actly as the differentials of the functions of only one variable, 
considering the other variables as constant. It remains, then, 
only to see how the total differential can be obtained. Before 
we proceed to this, observe that the differential of y =f (x) is 
expressed by dy = /' (x) dx. Now, the derivative j' (x) is a 

function of f(x). Suppose, for example, / (x) = ^2px, from 
which f f (x) = \J g- . A change of any of the three factors 

2, p, x contained in f (x) will be evidently attended by a cor- 
responding change in /' (x) ; and if the change be infinites!- 



DIFFEKENTIAL CALCULUS. 79 

mal, the change in/' (x) also will be infinitesimal. Thus, sup- 
posing, for example, that we make an infinitesimal change in 
the coefficient 2p, the derivative will be /' (x) ± <5, differing 
from /' (x) by the infinitesimal quantity <5. Hence to the 
equations 

y=f(x), dy=f(x)dx 

we may add another, 

dy, = if (*) ± s l *h 

corresponding to the same value of the independent x, when in 
the given function f(x) some other element besides x is sub- 
mitted to an infinitesimal change. JSTow, from the last equa- 
tion we have dy, = f f (x) dx ± 8 • dx. But 5 • dx is an infini- 
tesimal of the second order, therefore dy, = /' (x) dx = dy ; 
i. e., the differential of y = f(x) remains unchanged whether 
the other elements of the function do not change together 
with x or be they also submitted to an infinitesimal change. 
Let us now apply all this to 

differentiating first /x with regard to x, we have 

d x[ L =/(a? + dx, y, z) —f(x, y, z) ; 
differentiating then / (x + dx, y, z) by y, we shall obtain the 
same result as by differentiating the given fx by y ; i. e., 

dyP = /(# + dx, y + dy, z) —f(x -f dx, y, z). 
Lastly, taking the differential of / (x + dx, y -f dy, z) with 
regard to z, we shall again obtain the same result as by taking 
the differential of the given /x with regard to z; i. e., 

c? 2l u =f(x + dx, y + dy, z + dz) — f(% -f- dx, y + dy, z) ; 
adding now together these three differentials, we obtain 
d x p + dy\h -f d z \h — f (x + dx, y -\- dy, z + dz) — / (x, y, z) ; 

but the second member of this equation is the total differential 
d^; hence 

c?/x == d x {k -j- d y \k -f- d 2 ^. 

The same process is applicable to any number of variables, 
6 



80 PRINCIPLES OF INFINITESIMAL CALCULUS. 

therefore The total differential of a function of different varia- 
bles is equal to the sum of the partial differentials of the same 
functions. 

Applying it to the case of p = - — :_»— , which we have 

sin 2 

taken above as an example of a function of different variables, 
we have (V. and VI.) 

j log (y) 7 7 x j 7 » log (y) cos z , 

d x p = . wy efc, d> = — : — ■ dy, d z p = — 5-^f cfe; 

smz ' * y sm 2 sm 2 2 

hence 

<Z ^^ = ^ dx + ^dy- X ] °g ^ C0S S dz. 

sm 2 sin 2 3/ sm 2 snr 2 

XVII. Derivative functions. 

Recalling to mind what has been said (IX.) concerning 

derivative functions, it will be easily admitted that -|~, -~ 

dx dy ' 

-Ar- j are the partial derivative functions of a = f (x, y, 2) with 

az 

regard to x, y, 2. These same functions are also represented 

b y /'* Of j y> z \ fy ( x > V, z \ f* ( x , y> z )- Thus, 

daP = f'x (x, y, 2) dx, d y p = f' y (x, y, 2) dy, d z p =f' z (x, y, z) dz, 

and consequently the formula dy* — c^ + cTyjx + d,p may be 
represented also by 

dp = /'* fo y* *) ^ + A 0> y» *) ^ + f* fa y> z ) d h 

or by 

j d x p 7 C?^ 7 , ^M- 7 

off* = — — cte + -~- ay -\ — -- dz. 
dx dy dz 

Xay, since the partial differentials with regard to x, y, 2 are 
sufficiently indicated by their respective denominators, the signs 
x, y, z affixed to d may be and are ordinarily omitted. Repre- 
senting thus the differential of p more simply we have 

, dp j .dy. , dp 

dp— — dx -f -j- dy + — az. 
dx dy dz 



DIFFERENTIAL CALCULUS. 81 

In like manner, following the analogy of the differentials and 
derivatives of various orders of the functions of only one varia- 
ble, we represent by d 2 x \x, d 2 y ix, d 2 z ^, the differentials of the 
second order of fx relatively to x, y, and z, and by d x d y ^, d x d z p, or 
d x dyd z \h the differentials of fx, first with regard to y and then 
with regard to x, or first with regard to z and then with regard 
to x, or first with regard to z, then with regard to y and then 
with regard to x, etc., by succeeding differentiations. Also 
the corresponding derivatives will be represented by 

... , x ... , N , d 2 ^ d 2 v» d 2 & d 2 p , 

f' x {x,y,z),r> y {x, y, z), ete.,orby^ - rf - ¥ , ^, ^,. .. hence 

d 2 x \^ =f"x (%, y, z) dx 2 = -~ dx 2 , 
^2 

d \v< = S"v (?* y> z ) d v 2 = ^2 d tf> 



and 



,72 

«P«f* = f"z {x, y, z) dz" = -r-j dz, 

dz" 

d x d„\i> = 7 T dxdy, d x d z \h = — — — dxdz, . . . 
dxdy dxdz 



Whatever be the order kept in the successive differentiation 
of p, first with regard to x, for example, and then with regard 
to y, or vice versa, the result is the same, for from the differen- 
tials 

4f* =/ + dx, y, z) —f (x, y, z), 
d^==f(x, y + dy, z) —f (x, y, z), 
we obtain 

dydxit. =f(x + dx, y -f dy, z) — f(x, y -f dy, z) — f(x + dx, y, z) 
d x dyt*. =f(x + dx, y + dy, z) —f(x -f dx, y, z) —f(x, y + dy, z) 

+f(v>y> z )- 

Now the second members of these equations are identical; 

dyd x y* = d x d^. 



/ 

82 PRINCIPLES OF INFINITESIMAL CALCULUS. 

For example, take again jx = — -°3^. We shall have 

sin 2 

7/7 rJ ^°& (2/) dx dydx 

sin z ysmz 

y j. , xdy dxdy 

y sm z y sm z 
With the preceding observations, it is not difficult to find 
the formulas by means of which we may obtain the successive 
and total differentials of a function of different variables. Let, 
for example, jx = F (x, y) be a function containing two inde- 
pendent variables, x and y. We shall have for the differential 
of the first order 

d[x = d x ik + cLf* = — - dx -f — dy, 
ax dy 

and for the differential of the second order 

cP/x = d x (d x p + d y \i) + d y (d x \i> + d y \t). 

Now d x {d x \k + d y y) = d 2 x {x + d x d y n, and d y (d x p + d^f*) = fPyf* 
+ d x d y [k ; hence 

c? 2 a == d 2 x ^ -f eP«f* + 2d x d v p = — 9 die 2 -f — . cfy 2 -f- 2 — - — dxdy. 

ax-' ay dxdy 

If the given function F (x, y) be constantly equal to zero or 
to any constant quantity C, d^ = d 2 p == ; i. e., 

— dx + — c??/ = 0, 
cte dy 

d ^dx 2 + ^Jcfy 2 + 2 -^ da% = 0. 
dx 2 dy 2 dxdy 

But in this case one of the variables, y, for example, is func- 
tion of the other ; hence dy—^- dx, d 2 y — -^ dx 2 , and the first 

ax ax 

of the last two equations becomes—- dx -f -- . - - dx — 0. The 

dx dy ax 

second term of the second equation, or its equivalent d 2 y p. = 

dF' y (x, y) dy, in which dy is variable with y, is the differential 

of a product of two functions of y ; hence 



DIFFERENTIAL CALCULUS. 83 

d> = F", (x, y) df + F' y (x, y) cPy = g df + i?<Py 

_d 2 v .df , 2 dpdPy,, 
ay dx ay ax" 

This value substituted in the above second equation, together 

with that of dy in the third term, gives — —? dx 2 -f — ~ 9 dx 2 

dx" dy" dx" 

-{- — —4? dx 2 + 2 — — -f- dx 2 = 0, in which dx 2 is common 
dy dx" dxdy dx 

factor as dx is common factor of the corresponding preceding 

equation ; hence, in the supposition of y = / (x), we infer from 

the above equations 

dp. d^ dy 

dx dy dx ' 
(Fp dV df dp d~y , 2 _^V. ^ = 

Now (XV.) -j- or /' (x) = tg (to). Placing this value in the 

last equations, and F (x, y), or simply F, instead of fx, we obtain 
from the same 

f . dF , dF, . 

I d*+d/ s( } ' 

( } j * F cPF 2 dF «$ cPF _ 

t d? + d</ 2 tg {tX) + dy ■ dx* + Z dxliy- tg W - °- 

XVIII. Singular points of plane curves. 

We call singular those points of a curve which present some 
peculiarities inherent to the character of the curve. Such are 
the multiple points double, triple, etc., L e., those through which 
pass different branches of the curve, each having a different 
tangent. An example of this kind is represented by Fig. 21. 
Points of regress or cusps are likewise singular points. They 
are those in which a branch of the curve stops to begin, as it 
were, another branch, both branches having in the same point 
a common tangent, whether the two branches turn mutually 



84 PRINCIPLES OF INFINITESIMAL CALCULUS. 

tlieir convexity, as in the first example of Fig. 22, or one of 
them turns the convexity to the concavity of the other, as in 
the second example of the same figure. Isolated, or conjugate 
points, are also called singular points. They are entirely sepa- 
rated from the branches of the curve, although their co-ordi- 
nates fulfil the equation of the same curve. In treating of 
these classes only of singular points, we shall avail ourselves 
of the last formulas (D) of the preceding paragraph. 

Let F (x, y) = be the equation of a curve referred to 
orthogonal axes, and let (fa?) be the angle which the geomet- 
rical tangent of the point (x, y) forms with the axis of abscissas. 
The formulas (D) co-exist with F (x, y) = 0, and must be 
simultaneously verified for each point (x, y) of the curve. 
Is"oav the first (D) is verified either when both terms are equal 
but affected with opposite signs, or when each term is sepa- 
rately equal to zero. When the factor tg (tx) admits of differ- 
ent values, as in the case of multiple points, since for the same 

dF dF 
values of x and?/, the derivatives — - , — - do not change, if for 

ax ay ° 

one of the values of tg, (tx) the two terms mutually eliminate 
each other, they will not for another, unless we suppose both 
derivatives equal to zero. A similar observation is applicable 
to the case of tg (tx) imaginary, which happens for isolated 
points ; i. e., unless both derivatives be equal to zero, the equa- 
tion cannot be verified. Thus, if the equation F (x, y) = 
belongs to a curve which contains singular points, the co- 
ordinates of these points may be found among those which 

fulfil the equations 

dF dF 

In this supposition, the second (D) becomes 

mv d 2 F , 2d 2 F + h . , d 2 F ■ 2r < ' . 
(D 2 ) . . . -=-s + -r—r tg (to) + -7T to; 2 (tx) = 0, 
v ' dx 2 dxdy 6 v ; dy 2 B v ; 

from which, substituting the values of x and y deduced from 

d 2 F 
(D,), if -y-2 does not disappear, we obtain two values for 



DIFFERENTIAL CALCULUS. 85 

tg (tx), either real, and equal or unequal, or two imaginary 
values. In the first of which cases, the point (x, y) would be 
a point of regress, in the second, a double point, in the last, an 
isolated point. Should the terms of (D 2 ) disappear by the 
substitution of x and y obtained from (D,), in order to find 
whether the curve admits of any singular point it would be 
necessary to have recourse to derivatives of higher order. But 
let us see an example of each of the three cases just mentioned. 

Let, first, F (x, y) = y 2 + x* — x 2 = 0, from which — - = 4x s 

dF 

— 2x } — - = 2y. Hence, in the present case, the equations (D t ) 

become 

4x* — 2x = 0, 2y = 0, 

from which x = 0, y = 0, which fulfil the equation and belong 
to the point of the curve passing through the origin of the 
axes, which may be a singular point. To see if such be the 
case, let us take the partial derivatives of the second order, 

which are - 1 -^ = 12# 2 — 2, — — - = 0, —-? = 2, and placing in 
dx 2 dxdy dy 2 & 

them x = y = 0, -^- 9 = — 2, — — - = 0, -^—r — 2, hence the 
dx 2 dxdy dy 2 

equation (D 2 ) is, in this case, 

— 2 + 2tg 2 (to) = 0, 

and, consequently, tg (tx) — 1, tg (tx) = — 1. The origin of 
the co-ordinates is, therefore, a double point, and the branches 
of the curve have their tangent forming an angle of 45° on 
each side of the axis of abscissas. The form of the curve is 
similar to that of Fig. 21. 

Let, second, F (x, y) = ay 2 — x 3 = 0, from which -=- = 

dx 

r7F 

— Zx 2 y ~ = 2ay; hence for the equations (D,) 

— Sx 2 = 0, 2ay = 0, 



86 PRINCIPLES OF INFINITESIMAL CALCULUS. 

and, consequently, x = y = 0. Taking the partial derivatives 

of the second order, and making in them x = y = 0, we 

d 2 F d 2 F d 2 F 

find — -=■ = — Qx = 0, -zr—. — = 0, -r^— = la ; hence (D 2 ) 
aV ' dxdy d 2 y ' K 2J 

2a tg 2 (tx) = 0, 
and, consequently, tg (tx) = ± 0. Hence the point of the curve 
corresponding to the origin of the axes is a point of regress, 
and the branches of the curve have, in that point, the axis of 
abscissas for common tangent. This curve is called a cubic 
parabola, and is represented by the first Fig. 22. 

. dF 
Let, -third, F (x, y) = y 2 — x i -j- a 2 x 2 = 0, from which — 

ax 

= — 4x* -f- 2a 2 x, — - = 2y, and, consequently, for the equa- 
tions (D,) 

— 4x 3 + 2a 2 x = 0, 2y = 0; 

consequently in this case also x = y = 0. Taking now the 
partial derivatives of the second order and making in them 

rT 2 F r1 2 F 

x = y = 0, we obtain -— - = — 12# 2 + 2a 2 = 2a 2 , - 7 — — = 0, 
v ' dx 2 ' dxdy 

d 2 F 

— -s == 2. Thus, in this case, we have for (D 2 ) 

ay J 

2a 2 + 2 tg 2 (tx) = ; 



hence tg (tx) == ± a^ — 1. The point, therefore, correspond- 
ing to the origin of the axes, is an isolated point. 

XIX. Convexity and concavity — points of inflection. 

As the derivatives of a given equation F (x, y) offer a crite- 
rion to find out if the corresponding curve admits of any sin- 
gular point, so the derivatives of y =f(x) offer a criterion to 
determine whether the corresponding curve turns its convexity 
or its concavity to the axis of abscissas. 

Let (Fig. 23) M be a point of the curve represented by 
y = f(x) having AK, KM for its co-ordinates x and y, and let 



DIFFERENTIAL CALCULUS. 87 

KK' be an infinitesimal increment dx of x. We shall have 
AK' = x + dx, and the corresponding ordinate K'M' = 
f(x-\- dx). Represent by v and u the abscissas and ordinates 
of the tangent of the point M, and let u, be the ordinate cor- 
responding to the abscissa AK'. It is plain that from M for- 
ward, the curve will turn its concavity or its convexity to the 
axis of abscissas according as M'K — NK' is negative or posi- 
tive, that is, according as 

f(x + dx) — u, < or > 0. 

Now the equation of the tangent is (XV.) u — y = {v — x)f f (x), 
therefore u, — y = (x + dx — x) f (x) = /' (x) dx, or u, =f(x) 
-f /' (x) dx. 

Taylor's formula gives us, besides, (XI.) 

f(x + dx)=f (x) + f (x) dx + \f" (x) dx 2 + 
■ ./'" (x) dx + ... 



2-3 
Hence, subtracting from this the preceding, 

fix + dx) — u, = \f" (x) dx -f ^—xf rrf (x) dx + . . . 

Now the sign of the second number depends on that of the 
first term, or rather on that of the factor /" (x). Therefore, 
supposing that/" (x) does not vanish, f(x + dx) — u, will "be 
< or > when/" (x) is < or > 0. Therefore, if the deriva- 
tive of the second order of y = fix) is negative, the curve from 
M forward turns its concavity to the axis of abscissas, and if 
the same derivative results positive, the curve turns its con- 
vexity to that axis. If/" (x) vanishes, the criterion will be 
taken from/ 7 " (x), and if this also vanishes, from/ IV (#), etc. 

Should the branch of the curve from M toward the axis 
AY, change the bending of its curvature, the point M is then 
called a point of inflection', and the branch of the curve turns 
its convexity toward the axis of abscissas on one side of it, and 
on the other its concavity. The derivatives of the equation 



88 PEINCIPLES OF INFINITESIMAL CALCULUS. 

of the curve, which we shall continue to represent by y = f(x) y 
will reveal to us if M is a point of inflection, which is one 
of those that belong to the class of singular points. Let 
(Fig. 24) M be a point of inflection of M"MM', and TT'the 
tangent corresponding to M, which, without ceasing to be a 
tangent, must necessarily cut the curve in M. Taking KK' 
= KK" = dx, and representing as before by u, v the ordi- 
nates and abscissas of the tangent, by u, the ordinate corre- 
sponding to AK /r or AK r ; since the differences M'K r — N'K', 
M r/ K /; — N^K" must be necessarily affected by a different 
sign, the signs also of / (x + dx) — u, and / (x — dx) — u, 
must be different from each other. In order to see if and 
when this condition is verified, and consequently if the curve 
admits of one or more points of inflection, let us resume the 
equation already obtained above. 

/ {x + dx) — u, = \r (x) dx + JL/'" (») dx 3 + . . . 

from which, changing in it dx into — dx, 

/(* — *») - *, = if" (») dx* - ~f" (*) dx 3 + ... 

These equations show that the differences cannot be affected 
with opposite signs unless f /f (x) = 0. Hence, if the curve 
admits of any point of inflection, the co-ordinates of that point 
must fulfil the equation f ff (x) = 0, and, consequently, vice 
versa, the co-ordinates of the points of inflection must be 
found among those which fulfil the same equation. It is not 
sufficient, however, that real co-ordinates x m , y m fulfil the equa- 
tion f" (x) = to enable us to infer that the curve admits of 
points of inflection, if the same co-ordinates x m , y m annul the 
derivatives of higher orders, except when the first derivative 
which is not annulled is of an uneven order, 3d, 5th, etc. 
When, therefore, the equation y =f (x) is such that for certain 
real co-ordinates x,„, y m , the derivative f" (x) becomes zero, 
and the first of the subsequent derivatives which does not 
vanish is of an uneven order, the curve admits of as many 



DIFFERENTIAL CALCULUS. 89 

points of inflection as there are pairs of co-ordinates x m , y m for 
which the said conditions are verified. 

Let us take, for example, the transcendental curve repre- 
sented by the equation y = /(#) = sin x, from which (IX. and 
VIII. in.)/' (x) = cos x,f" (x) = — sin x, f" (x) = — cos x, 
f IV (x) == sin x, etc. The first condition to be verified is that 
j" (x) = be resolved with real values x m of x. Now x m = 
0, = it, = 2nt, = 4*y == . . . are all real roots of /" (x) = 0. 
The second condition is that /'" (x) or/ v (x), etc., is the first 
of the derivatives which does not vanish when x m is placed in 
them. But, in our case, f" r (x) = — cos x = — 1, + 1, — 1, 
+ 1, — etc., when x = 0, = *, = 2*, = . . . Hence the curve 
represented by y = sin x admits of an infinite number of 
points of inflection. The form of this curve is j>artly indi- 
cated by Fig. 25. Its tangent, at the origin of the axis, bisects 
the angle formed by the same axis : for tg (tx) =/' (x) = cos x, 
and with x = 0, cos x = 1 ; hence tg (tx) = 1 = tg (45°). 



PART II. 

INTEGEAL CALCULUS. 

XX. Indefinite Integrals. 

The object of integral calculus is opposite to that of differ- 
ential. It consists in finding the function from which a given 
differential has been obtained. Integral and differential are 
correlative. Thus, as adx is the differential of ax, so ax is the 
integral of adx. The integral of a given differential is desig- 
nated by affixing to it the symbol f, which signifies sum, as the 
letter d, adopted in differential calculus, signifies difference. 
f adx signifying the same thing as ax, we may write the 

equation 

(i) . . . f adx = ax, 

the first member of which indicates, the second expresses, the 
integral of adx, and the equation is read Integral of adx is 
equal to ax. 

Let us here observe two things: first, that as (IX.) f (x) dx 
represents the differential of any function of x, f (x), and as 
df(x) and /' (x) dx signify the same thing, we shall have 
fdf{x) == ff (x) dx = f (x) ; i. e., the integral of a differential 
only indicated, is obtained by suppressing both signs f and d. 
Secondly, that as, C being a constant, d (/ (x) + C) = (V.) 
df(x) equal, in both cases,/ 7 (x) dx; ff (x) dx may be given 
by fix) + C, as well as simply byf(x) ; i. e., 

f (x) + C is called complete, and / (x) incomplete, integral of 
f (x) dx. In both cases, however, the integral is called indefi- 
nite, for the reason to be given in a following number. 

90 



INTEGKAL CALCULUS. 91 

XXI. General theorems. 

Resuming again / (x) and its differential f (x) dx, we shall 
have besides df(x) =f (x) dx andy/' (x) dx —fdf{x) —f(%), 
also, representing by a a constant, 

aff (x) dx = afdf(x) == af(x); 

but from df (x) = f (x) dx we have also adf (x) = af (x) dx 
and adf(x) = daf(x) ; hence 

faf (x) dx =fdaf(x) = af(x). 

Therefore f a f f { x ) dx — aff (x) dx ; i. e., 

1st. Constant factors of a given differential to be integrated 
may be placed outside the sign of integration. 

We know (VII. i.) that d [(F (x) +f(x) + <p (a?)] = d¥.(x) 
+ df(x) + dcp(x); hence also f [d F (a?) + df(x) + d(p(x)] = 
fd [F (a) + f(x) + <p (a;)] ; but /d [F (a?) +f{x) + <p (a?)] = 
F (a .) + jF («) + ? (*) and /d F (x) = F (x),fdf(x) =f(x) } 
fd cp (x) = <p (a?), therefore 
ytdF(aj) + df(x) + d*(x)] =fd¥(x) +fdf(x) +/d'i(x) } 

i. e., 

2d. The integral of the sum of different differentials is equal 
to the sum of the integrals of each term. Thus (V. n., and VI. 
III. and i.) 

fladx — b cos xdx + c — ) = ax — b sin x + e log (#). 

We have (V. IV.) dx m + x = (m + 1) a^a?. Therefore 

/(m + 1) # m efo; = fdx m + 1 = x m + \ Now /(m + 1) a; m cfo = 

(m + l)fx m dx, therefore 

x m + 1 

fx m dx = ; i. e., 

J m + 1 

3d. The integral of x m dx is obtained by suppressing dx, 
adding 1 fo £/ie exponent m, anc£ dividing x m + 1 63/ m + 1. 
Thus, for example, 

fZtfdx = Sfx 2 dx = x 3 , 



92 PRINCIPLES OF INFINITESIMAL CALCULUS. 

— ^- = a fx ax = , 

x 2 J x 



f^x 2 ' dx —fx 3 dx = — >/x 5 . 

o 

XXII. Immediate integration ; integration by substitution ; 
integration by parts. 

These different methods of integration are used, now one, 
now another, according as circumstances may suggest. 

I. When the given differential is such as to show imme- 
diately the function from which it has been obtained, as in the 
cases examined, Nos. V. and VI., the integration is ob- 
tained immediately without having recourse to any rule, and 
on this account, the integration is called immediate. Thus, for 
example, we know (VI. IV.) that sin xdx is the differential of 
— cos x ; hence we conclude immediately 

f sin xdx = — cos x. 
In like manner (VI. n.) we obtain immediately 

J ' a x l (a) dx = a x ; 

and since a x dx = ——. a x l (a) dx. also 
1(a) 

fa x dx = — -r a x . 
J I (a) 

Also (VI. vil, viii., ix.) 

/ax , , \ r tLX . . 

— — = arc (sin = x), J , = arc (cos = x). 

Si—a? </! 



x 2 



f -i o = arc (tff = x). 

J 1 + x 2 v 8 ; 

II. When the given differential does not show from its form 
the function from which it has been obtained, it is then by 
means of substitution so modified as to take one of the 
known forms, and the integral is then found. This method 
is accordingly called method by substitution. Let, for example, 

x (^a 2 — x 2 )dx be a given differential. Make a 2 — x 2 = z f 



INTEGRAL CALCULUS. 93 

fa 

and consequently dx = — --. Thus f x (y a 2 — x 2 ) dx = 

Ax 

3 3^ 

,-dz — z 2 z 2 

—fx (^)g = —\fz 2 dz = — 1-3- = 3- • Therefore 

2 

3 



fx {^a 2 — x 2 ) dx — J (a 2 — x 2 ) 2 . 

dx x 

But let =t 7 — ' be the given differential. Make — = z : 

(s/ a 2 —x 2 ) a 

dx 
i. e. ; a; = az, consequently dx = adz, we shall have/" ± 



^a 2 — x 2 



= / + _ ac?0 _ = y± — ■— . Now (VI. Vii.) -f 



= cZ arc (sin = z) and = d arc (cos = 2;). Therefore 

y.l — z 2 

/ — - - = arc ( sin = — ) , 
J V a 2 —x 2 K a>' 

r dx , x x 

/ = arc ( cos = — . 

«./ 2 2 \ n 

Let finally the given differentials be cos x sin 2 xdx and 
sin x cos 2 #da\ Make for the first of them sin x = z, and con- 
sequently cos xdx = dz, for the second cos x — z, and conse- 
quently — sin xdx = dz ; we shall have 

f cos x sin 2 #cfc = yVcfe = \£ = 



sin 3 # 



f sin a; cos 2 xdx = / — ■ z 2 dz — — Js 3 = — 



COS 3 07 



3 



Changing in these formulas x into \x, 



sin 3 ia; „ . , o , -,, cos 3 ia? 



f cos J# sin 2 J# d\x — — — — , /sin \x cos 2 Ja; df # == 

But dja; = \dx, and the constant coefficient \ being brought 
out of the integral sign, the same equations will be changed 
into the following . 
/cos \x sin 2 \xdx — § sin 3 \x, /sin \x cos 2 \xdx = — § cos 3 } x. 



94 PRINCIPLES OF INFINITESIMAL CALCULUS. 

III. Integration by parts is effected when, instead of inte- 
grating the given function, the same is resolved into two parts, 
each one of which is integrated separately. This method is 
based on a formula which we establish as follows : Represent- 
ing by y and z two functions of the same variable x, we have 
(VII. II.) d (y z) — zdy -f ydz ; hence 

ydz = d(yz) — zdy ; 
i. e., the product ydz of a finite by a differential function of 
x is resolvable into two differential terms, the integration of 
which, if more conveniently obtained, may be taken instead 
of that of ydz, to which it is equal. Now the last equation 
gives us 

fydz = yz— /zdy; I e., 

The integral of the product of two functions of the same variable, 
one finite and the other differential, is obtained by tahing from 
the product of the first function by the integral of the second, the 
integral of the product of the second integrated by the differential 
of the first. 



Let, for example, ^a 2 — x 2 x dx or its equal ( \J ~- — 1 ) xdx 
be a given function to be integrated; we shall have 




a 2 



fVtf — x 2 X dx =/(V - 2 — 1) xdx = — (V. IV.) 

/(\/^)4 ■ 

The latter member is represented by the above general for- 
mula, hence 



=(v / f-i)?-4-^-i)- 1 ^ 



-.-r r-r-r N *& d i i **Gi X , .Ztt OlX TT 

ISTowPv. III.) d- 9 — — —. dx- — r- dx = — ., . Hence 

N ' or x x x 



INTEGRAL CALCULUS. 95 

a 2 dx 



As/^)4-(V^)i+f 



Wi- 



X ' Z K m X 



2 J ^ a 2 — x 2 ' 
but we have found above /* " — = arc (sin = — ), there- 

J -v/7.2 ™2 V a I 



v^a 2 — x 2 a 

fore 

x 

a 



f(^a 2 — x 2 ) dx = \ [x\/d z —x 2 + a 2 arc (sin = — )]. 



3 



Let, for another example, (a 2 — x 2 ) 2 dx be the given fume- 
s' 

tion to be integrated. Observe first that (a 2 — x 2 ) 2 — (a 2 — a* 2 ) X 
(Va 2 — x 2 ) = a 2 ^(^T—x 1 ) — x 2 ^a 2 —x 2 , also x 2 ^a 2 — x 2 
= x X x v' 'a 2 — a 2 ; hence 

_3_ 

(a 2 — a 2 ) 2 dx= a 2 ^{a 2 — x 2 ) dx — x X x ^JaF — x 2 ) dx. 
Now x v/(a 2 — x 2 ) dx — — d J (a 2 — x 2 ) 2 ; therefore 

j$_ _3_ 

/( a 2 _ a 2 ) 2 gfe = ^/^(a 2 — a 2 ) cfo + |/a 'd(a 2 — x 2 ) 2 ; 

the last integral of this equation is again represented by fydx, 

1 A 

and consequently = x (a 2 — x 2 ) 2 — f(a 2 — x 2 ) 2 dx. There- 
fore, 

f{a 2 — x 2 ) 2 dx = a 2 f^{a l — x 2 ) dx + ±x (a 2 — x 2 ) 2 — 

jy"(tt 2 — x 2 ) 2 cfc, 

_3 

from which, taking the common factor /(a 2 — x 2 ) 2 dx of the 
first and last term, alone in the first member, 

3 _3 

/{a 2 — x 2 ) 2 dx = ja 2 yV(a 2 — x 2 ) dx + \x {a 2 — x 2 ) 2 • 
but, from the preceding example, 

f^{a 2 — x 2 ) dx = %[x ^{a 2 — x 2 ) + a 2 arc • sin (= — ) ]; 
hence, finally, 



96 



PRINCIPLES OF INFINITESIMAL CALCULUS. 



f(a 2 — x 2 ) 2 dx — %a 2 x ^\a 2 — x 2 ) + fa 4 arc * sin (= — ) 



+ J x (a 2 — x 2 ) 2 . 

The same method of integration by parts is applicable 
immediately to the differentials x ' sin xdx, x ' cos xdx ; i. e., 
observing that (VI. in. and iv.) sin xdx — d — cos x, cos xdx 
= d sin x, 

(fx ' sin xdx = fx ' d — cos x — — x cos x — 

■ f— cos xdx = sin x — x cos x. 

(1st.) \ . 

fx ' cos xdx — fx ' d sin x = x sin x — f sin xdx 

= cos x + x sill x. 
From these two equations we infer the following : 
f2x * sin 2xd'2x = sin 2x — 2x cos 2x, f\x sin \xd\x = sin \x 

y 2a; " cos 2xd2x = cos 2a; + 2a; sin 2a;, f Ja; cos \xd\x = cos Ja; 

+ Ja; sin Ja;, 
which, bringing out of the differential and integral signs the 
constant coefficients 2 and \ , are easily changed into 

y# sin 2a;da; == J sin 2a; — \x cos 2a;, 
y*a; cos 2xdx — J cos 2a; + \x sin 2a;, 
y*a; sin \xdx = 4 sin Ja; — 2a; cos Jo?, 
fx cos Ja;c?a; — 4 cos \x + 2a; sin Ja\ 

Let, for a last example, sin 3 a;cfo;, cos 3 xdx be the differen- 
tials to be integrated. First, they may be resolved into two 
factors as follows : sin 2 x ' sin xdx, cos 2 x ' cos xdx ; but sin xdx 
= d — cos x, and cos xdx == d sin a;. Thus 
/sin 3 xdx =ysin 2 x'd — cos x = — sin 2 x cos x — f- — cos x d sin 2 x, 

= — sin 2 x cos x + 2y"cos x sin x 

d sin x)y 
= — sin 2 x cos x + 2y*cos 2 a; sin a;c?a', 
= — sin 2 x cos x -f 

2f(l — sin 2 x) sin a;e?a;. 
Now f (1 — sin 2 a;) sin a?<ia; = y (sin a;cfe — sin 3 xdx) = 



(2d.) { 



XNTEGKAL CALCULUS. 97 

■ — cos x — f sin 3 xdx; therefore/* sin 3 xdx = — sin 2 x cos x 

— 2 cos x — 2y*sin 3 xdx. Hence 

/* sin 3 xdx = — J- sin 2 x cos x — § cos x. 
In like manner we shall have 

y*cos 3 xdx = /"cos 2 x ' d sin x = cos 2 x sin x — /"sin x ' d cos 2 x 9 
and with a process altogether like the preceding we find 

/"cos 3 xdx = J cos 2 x sin x + § sin #. 
These and the preceding integrals become completed by adding 
any arbitrary constant, C, to them. 

XXIII. Definite or limited integrals. 
Let C be any arbitrary constant and y —f (x) + C be any 
continual function of x, we shall have dy — /' (x) dx. Now 
f(x) may be taken, as it is in reality, for the sum of infinites- 
imal elements dy as many in number as there are infinitesimal 
elements dx in the x of f(x). Calling now x the particular 
value of x which makes/ (x) — — C, we shall have/ (x ) = y 

— = C, and consequently 

y=f( x )—f( x o), 

in which y, which is the integral of/' (x) dx, represents the 
sum of as many infinitesimal elements dy as are the infinitesi- 
mals dx contained in the difference x — x ; for the sum of those 
elements which would make /(a?) =f(x Q ) is destroyed hj f(x Q ). 
But so long as x has no fixed value, y or ff r (x) dx remains 
undetermined. It will become determined when x receives 
a particular value, for instance, x in . Now to designate that 
the y corresponding to the integral of /' (x) dx represents the 
sum corresponding to x — x Q or to x m — x , we affix to the sign 

f the two values of the variable as follows : f or f\ n \ and 

Xq Xq 

we consequently write : 



98 PRINCIPLES OF INFINITESIMAL CALCULUS. 

in both of which we must remember /(a? ) replaces any arbi- 
trary constant C. 

Further to illustrate what has been said above, make x + 

Ll/Ju — Xf, «£/ i LIX —~ X 2 * X 2 ~i aX ~~~ Xq« • • • iC;/| _ ]^ — J— (XX 



3, . . . *v /n — j^ | yi«v a? m , 



we shall have 

f(x + &?) —f(x ) = /' (x ) dx = dy =f(x,) — f(x ), 
f (x, + dx) —f (x) = f (x,) dx = dy, = f{x 2 ) —/(a?,), 
/(a* + cfo) —f(x 2 ) =f (x 2 ) dx = dy 2 =f{x 3 ) —f(x 2 ), 

f(x m _ i + cfo) — /(a? m _ i) =/' (a? OT _ i) dx == cft/ m _ ! =f(x m ) 

hence 

<fyo + <%' + • • • + c ty>n-i =f{%m) —f{x ) ; 

but dy -j- cfa/, + . . . + dy m _i is the sum of all the elements 
of y from y = y to y = y m , corresponding to those of x from 
x = Xq to a; = x m ; y , as stated above, is = ; hence the sum 
of all the elements is y m or the difference f(x m ) — f(x ). Now 

y m = ff f (x m ) dx, which is represented by/" m f (x) dx; there- 
to 

fore the equation (il), in which x m represents any determined 
value ofx, which changed into another x n will give f n f f (a?) dx 
= / (x n ) — / (x ), and consequently 

f*~f (*) dx -f*"f (x) dx =f(x m ) -f(xi). 

But/(a? m ) = y m ,f(%n)=yn ; hence/ (x m ) — f(x n ) = y m —y n , 
which contains all the infinitesimal elements dy corresponding 
to those of x in the difference x m — x n . Hence f(x m ) — f(x n ) 
is the limited integral of f (x) dx between x m and x n) and ac- 
cordingly represented by/"* " l f f (x) dx ; i. e., 

x n 

(ill.) f X '"f (x) dx =f{x m ) -f(x„). 

We infer, therefore, from what precedes : 

1st. The integral of a differential expression F (x) dx taken 



INTEGRAL CALCULUS. 99 

between two determined limits, is the sum of all the values which 
F (x) dx receives by the infinitesimal variations of x from one 
limit to the other. 

2d. The integral of the same expression is given by the differ- 
ence of the values which its indefinite integral, abstraction being 
made from the constant, receives when we substitute in it the 
values of x corresponding to the two limits. 

To illustrate these theorems, let, for example, RR' (Fig. 26) 
be a parallel to the axis of abscissas OX of the orthogonal 
system YOX, and consequently perpendicular to OY. The 
ordinates of the different points of RR' are necessarily all 
equal to the constant segment OR, which we shall call h; 
y therefore does not vary with x, but the area limited by OR, 
RR', OX and the ordinate of a point of RR', varies with the 
abscissa of that point. This area, therefore, is a function of 
x, which we may represent by y. Now the same area is the 
product of the constant ordinate h by the variable abscissa. 

Hence 

y = hx ; 

and since, in our present supposition with x = 0, y also is 
equal to zero, no constant is added to hx, or the constant, in 
this case, is zero. 

Let now KK' = dx, we shall have dy — KM', and the in- 
tegral of KM' (— hdx) is the area KR. But if in the equa- 
tion y = hx, instead of x = OK we would take x = OB, the 
integral of hdx would be the area BR ; in each case, however, 
the area is evidently equal to the sum of as many infinitesimal 
areas KM' = dy as there are infinitesimal elements dx in x = 
OK or = OB. Now OK and OB are any two abscissas, and 
to distinguish one from the other we may call the first x m and 
the second x a . x m — x n is then the segment BK of the axis 
of abscissas between the ordinates y m , y n , and the area KA or 
(x m — x n ) his = x m h — xji, which is the sum of as many elements 
dy as there are infinitesimal elements dx in the segment x m — x n 
= BK, i. e. the integral of KM' no less than KR, with this dif- 



100 PKINCIPLES OF INFINITESIMAL CALCULUS. 

ference, that the latter area is taken from x = to x = n, and 
accordingly designated by J\ m kdx; the former is taken from 

x = x n to x — x m , and consequently expressed by f\ m hdx; and 

x n 

as in the first case f~ ni kdx = hx m , so in the second 

Xn 

which corresponds exactly with the preceding theorems. 

We may now proceed to apply the last theorem to some par- 
ticular functions. But first let us observe that an altogether 
indefinite integral may be expressed by the integral limited 
by one term. For taking the integral of f (x) dx from x to 

x, we have/** f (x) dx =f(x) — f(%o) and consequently 

Xq 

dfvf ( x ) dx = d f( x ) ~ d f( x o) = f ( x ) dx > 
and 

//' Op) dx =/• a/*f (*) ^ =/? f (x) dx ■ 

but 

//' (*) dx =f(x) + C and /fy («0 dx = f(x) -f{x ), 

in which — ffoo) represents any arbitrary quantity C; hence 

ff (x) dx -r f (a?) dx, 

x being that particular value of x which renders fix) — — C. 
Coming now to the applications, let 

X <T + X" 

dx 
be the given differential functions. — = dx or — is (VI. I.) 

° x" x" 

the differential of I (x), therefore the indefinite integral 



INTEGRAL CALCULUS. 101 

of — dx is I (x) + C ; hence the integral of the same function 

from x = x Q to any value of x of this variable is 

r x 1 

x o x 



f —dx= log (x) — log Oo). 



dx 
The second function, or its equal -^ — r 2 , is (VI. ix.) the 

Cv — ^ — X 



differential of — arc (tg =— ). Therefore its indefinite in- 

Cv Co 

tegral is — arc (tg = —J -f- C ; hence the integral of the same 



function from x = to x — a is 

f ?TP & = « ar ° (tS = 1} ~ I ar ° (% = 0) - 

Now arc (tg = 1) = 45° = — , arc (tg = 0) = 0; therefore 

r a 1 dx - « 
J d 2 —x 2 4a 

x m "^ ■*■ 

The last function is (V. IV.) the differential of -; hence 

m + 1 

x m + 1 
its indefinite integral is - + C, and consequently its defi- 
nite integral from x = to x = 1 is 

f x m dx 



m+1 

XXIV. Differentials of an are, of an area terminated by an 
arc, of a sector, and their corresponding integrals. 

i. Let (Fig. 27) the plane curve CC be referred to the 
orthogonal axes OX, OY, and let tf be an arc of it taken from 
A, a determined point whose co-ordinates are x Q , y , to another 
point M variable and whose co-ordinates are any two x and y. 
Let also y ~f(x) be the equation of the curve and KK' an 
infinitesimal increment dx of x. Drawing the ordinate K'M', 



102 PRINCIPLES OF INFINITESIMAL CALCULUS. 

and from M, MD parallel to OX, the sides MD, DM'of the 
infinitesimal right-angled triangle MM'D will respectively be 
equal to dx and dy. The infinitesimal arc MM' = da, which 
may be regarded as coinciding with the chord, will be equal 

to ^ dx 2 + dy 2 = t\J 1 + ~- 2 ) dx, and since ~ = j' {x), the 

\a>JU \X/JC 

differential of the arc d is 
and consequently the integral 

'/* Ki+/' 2 (*)) dx = * 

taken from the abscissa x — x to any other x, gives the recti- 
linear measure of rf for any plane curve ; hence the last formula 
answers the purpose of rectifying plane curves. 

Let, for example, AM (Fig. 28) be the cycloidal arc tf taken 
from the vertex where is the origin of the axes, to the point 
M whose co-ordinates are x and y. The equation of the cy- 
cloid referred to the rectangular axes and as already determined 

(XXVII, A. G.) isy=c • arc (sin = - 2eX — - -) + ^2 



-CX — or 
G 



in which 2c represents the diameter act' of the generating cir- 
cle. Taking the differential of this equation, and to simpli- 
fy the operation make ^2cx — x 2 = z, we shall have, first, 

y = c * arc (sin = — J -f- z. Therefore (YI.vn.) dy=e 



d^- 



/!=? 



\ 

* c* 

4- dz == (—■ rr ; + 1) dz* But z = ^2cx — x 2 : hence z 2 — 

\v/ c 2 — z 2 i 

2cx — x 2 and dz = - — . Therefore 

^2cx — x 2 

dy= ( - _ = + 1) -£=£= dx 

v c 2 — 2cx + x 2 ' v2cx — x 2 



INTEGRAL CALCULUS. 103 

= ( + 1) -} J — } dx 

^c — x ' V2cx — x l 

2c — x , 2c — x j 

dx = __ ax ; 



*S 2cx — x 2 *fx ^2o — x 

finally, 



dy=\f 






x 
and consequently /' (x) or 



dx V —J— 
Hence for the cycloid the arc <r is given by 



/" \/ — . dx. 
J x» v x 

But \/^ (& = ^2o • -&, and %. = d2v^; hence 

f\/ 2 ~'dx =r^2c'2d^= 2^¥c/d'^ r x = 2<S2e~'~x~; 
% a? 

therefore, from the preceding number, 

j.x>f2e dx = 2 y2c^_ 2v^2c^. 

^0 v X 

Taking x = or the abscissa of the origin A of the axes, and 
for x the abscissa AK corresponding to the point M, so that 
<r = AM, we shall obtain 

AM = 2^2-c-AK. 
Xow 2c is the diameter of the generating circle, and the chord 
AC which joins the extremity A of the diameter with the 
point C of the circle met by the ordinate KM, is mean geo- 
metrical proportional between the diameter and AK ; there- 
fore ^2 ' c • AK = AC ; hence the arc AM of the cycloid is 
the double of the chord of the generating circle joining the 
origin of the axes with the point met by the ordinate of M. 
But if AK should become equal to the diameter 2c, we would 
obtain AB = 4c ; i. e., the rectilinear length of the cycloidal 



104 PRINCIPLES OF INFINITESIMAL CALCULUS. 

arc from the origin of the axes to the base is twice the diam- 
eter of the generating circle ; hence the whole length of the 
cycloidal line is four times the same diameter. 

II. Let, secondly, a be the area of ABKM (Fig. 29), termi- 
nated by an arc tf of the plane curve CC, by the ordinates y , 
y of the extreme points of the same arc, and by the difference 
x — x Q of the corresponding abscissas. Taking, as before, KK' 
= dx y the infinitesimal area between the ordinates of M and M', 
dx and dd, is the differential of a, and consequently MKK / M / = 
(Za, MM', as infinitesimal, coincides with the chord ; hence da 
may be regarded as a trapezoid whose height is dx y and y, y + 
dy the parallel bases, therefore da = 1 (y + y -f dy) dx = ydx 
+ \dydx. Neglecting the second term as an infinitesimal of 
the second order, we shall have 

da. = ydx y 

and representing as usually by y =f{x) the equation of the 
curve referred to the orthogonal axes OX ■ OY, also 

da = f (x) dx. 

Now the integral of this function, taken from x to x, gives us 
a; i. e., 

Let, for example, the given curve be the ellipse referred to the 
axes of the curve, and whose equation is (XII., A. G.) y — 



— \S a 2 — x 2 , we shall have 



a 

a = f — Va 2 — x 2 ' dx = — / >/ a 2 — x 2 dx. 
x o a a x o 

Now (XXII. in.) 

/(yd 1 — x 2 ) ' dx — J [x^a 2 — x 2 -f a 2 * arc (sin = — )]• 

Taking x Q = this integral becomes zero, and taking x = a it 
becomes Ja 2 * — ; hence 

A 

a — f — •Sd 1 — X 2 = ^nab. 
J a 



INTEGRAL CALCULUS. 105 

But the area between the limits x = 0, x = a, is the fourth 
part of the area of the ellipse ; hence, calling A the area of the 
whole ellipse, <rtab will represent this area; or, since (XI. (2), 

A. G.) ab = a 2 ^T^7, 

A = irab = *a 2 ^l — e 2 . 

Let another example be taken from the cycloid, which 
(XXVIL, A. G.) is represented by the equations 
x = c (1 — cos w), y = c (w + sin u>) ; 
from which dx = c sin udu and ydx = c 2 w sin wc?w -f- c 2 sin 2 wc?w 

= (Trig. § 18 (g')) c 2 u sin wcta + c 2 ( ) du. T>ut ydx 

is the differential da of the area ; hence the area of the semi- 
cycloid, which corresponds to the definite integral of ydx taken 
from x = to x = 2c, is given by 

r 2(3 7 /2c r , . 7 , c 2 7 c 2 COS 2w , -i 
« =/ 2/^ = A L c w sm wc? w + 2 c?w ~ 2 — J ; 

but (XXI. ii.) the last member is resolvable as follows : 

9 r 2e . , , c 2 ,2c , c 2 r 2e n 7 

c / « w sm acta -f -- / ^ aw / cos 2waw. 

y 2 2 

Now with x = also u = ; with a; = 2c, w = ie ; hence 

2c * 

f w sin wdw = y* w sin wc?w = (XXII. Ill, 1st.) ic : 

/ du = f' du = ir f cos 2wc?u = /" i cos-2wd 2w = 
y ,y y 2 

J/* cos 2w <Z 2w = (VI. in.) i sin 2* =0 ; 
therefore 

o 

A 

Now a represents the area of the semicycloid. The area, there- 
fore, of the whole cycloid is 

2a = 3«rc 2 ; 

i. e., three times the area of the generating circle. 

If the axes, instead of being orthogonal, form any angle &, 



106 PRINCIPLES OF INFINITESIMAL CALCULUS. 

and the area AEHM = a be terminated by the oblique co- 
ordinates AE, MH (= y) ; the infinitesimal increment MHH'M' 
= da, corresponding to the infinitesimal increment dx of the 
abscissa OH, may be considered as a parallelogram, neglecting 
the triangle MM/D as an infinitesimal of the second order. 
Now the area of this parallelogram is HH 7 # MK; and HH' = 
dx, MK *= MH • sin d = y ' sin 6 j therefore 

da = y ' sin 6 * dx. 

x 
Thus a = sin 6 f ydx ; 

Xq 

which is a formula more general than the preceding to obtain 
the area of a plane surface terminated by a curved line. 

in. Let O (Fig. 30) be the origin of the axes OX, OY, to 
which the plane curve CC r is referred, and the pole of the 
polar co-ordinates, having OX for polar axis. Taking A for 
an invariable point, let x Q , y Q be the rectilinear co-ordinates 
OB, BA of that point, and p , w the polar co-ordinates OA and 
AOX of the same point. Let also x, y, and p, w be the recti- 
linear and polar co-ordinates of the movable point M. The 
curvilinear sector AOM, which, we shall represent by a! , in- 
creases by diminishing w. The same sector is also equal to 

AOB + ABKM — OMK = ^ + a — X l ; hence (VII. I. 

and II.) do! = d -f° + da — d ^ = da - 2*+£^. But da 

A A -j 

= ydx ; hence da' — - — - — * . But if we suppose a and w 

to increase together, taking namely the variations in a retro- 
grade order, the signs then of dx and dy will be changed, thus 

, . . ydx — xdy 
do! = =b £ __ * ; 

in which the upper sign is taken in the first, and the lower sign 
in the second supposition. 

To express the same differential by means of polar co-or- 



INTEGRAL CALCULUS. 107 

dinates, let us take the well-known formulas x = p cos w, y = 
p sin w, from which dx = cos w dp — p sin udu, dy = sin wcfy -f- 
p cos udfw. Hence 

xdy = p cos w sin w dp + p 2 cos 2 wow, 
ydx = p sin w cos wc?p — p 2 sin 2 wdw, 
therefore 

2/cfo — ^c7?/ = — p 2 (sin 2 w -f- cos 2 w) c?w = — p 2 aw ; 

hence 

da! — ± J-p 2 aw ; 

in which the upper sign is taken when a and w increase or di- 
minish together. The area of the sector or a' is therefore given 
by the second, or by the third member of the following equation : 

of = ± l y i/cfe — #ay = =fc J f p 2 eta. 

Let, for example, the circle be the curve in which the sector 
is taken from w = 0°, to w = 360° ; p in this case becomes a 
constant if the pole be taken in the centre, as we suppose it to 
be, and equal to the radius r ; hence 

a! = 4?r / d'ji = *r , 
as we know from geometry. 

XXV. Circular curvature ; osculatory circle and, radius of 
curvature of a 'plane curve. 

Representing by r the radius of a circle in contact with a 
straight line, the same circle approaches to, or recedes from, the 
tangent according as the radius r increases or decreases. In 
other words, the curvature of the circle varies with the radius, 
but reciprocally; it increases, namely, or decreases with the 

ratio — , which ratio may consequently be taken to represent 

the curvature of the circle having r for radius. The curva- 
ture of the circle being the same everywhere, it will be repre- 
sented everywhere by — . Now any plane curve may be con- 



108 PKINCIPLES OF INFINITESIMAL CALCULUS. 

sidered as composed of infinitesimal circular elements, and the 
circle corresponding to each element is called the osculatory 
circle of that element ; and consequently, r being the radius of 

the circle, — gives the measure of the curvature of the same 
r 

element. But the curvature of plane curves different from the 

circle is different at different points, consequently — is a varia- 

r 

ble quantity for these curves. 

Let (Fig. 31) CAC be any plane curve whose equation is 
y =f{%)> and let MAM' be one of its infinitesimal elements 
bisected in A. To determine the radius of the osculatory cir- 
cle of this element, observe, first, that the tangent of the curve 
corresponding to the middle point of AM coincides with the 
element AM; and the tangent corresponding to the middle 
point of AM' coincides with the element AM'. Let Tt, T't' 
be the two tangents, and m, m! the points of contact. Call 
(tx), (t'x) the angles which the same tangents form with the 
axis of abscissas, and (#') the infinitesimal angle which they 
form one with the other. Now the perpendiculars mT>, m'D to 
the tangents meet in the centre of the osculatory circle, corre- 
sponding to MAM', and MD is consequently the radius of this 
circle. Representing by dcf, AM = AM' = mm', the quadri- 
lateral mDm'A gives us 

roDwi' = 180° — mAm' = (#'). 
Therefore, taking the arc corresponding to (tt f ), in the circle 
having 1 for radius, and calling the arc also (ttf), we have 

(ttf):d(f::l:r; 

. da 

hence r= ( — } . 

Bat (ttf) = (tx) — {t'x) = — \_(tfx) — (tx)] = — d {tx) ; and since 

(XV.) tg (tx) = f (&), tx = arc (tg = /' (x)) ; also (XXIV. I.) 

i 
da = [1 -f f' 2 (x)~] 2 dx ; therefore 

[i + frfi = , VI IX j _ Li+riMT . 

r ~ d are (tg = /'(*)) lV ' '' f"(x) 



INTEGKAL CALCULUS. 109 

This equation gives the length of the radius of the oscilla- 
tory circle, or radius of curvature corresponding to any point 
(a?, y) of any curve, in a form easily applicable to particular 
cases. 

Let us take, for example, the parabola for which we have 

f(x) = ^2px, and consequently /' (x) = ^7= = jj\ = --, 

T) T (x) 7) 

f" (x) = — rj V ^= — -z- Hence the radius of curvature 

/ 0*0 y 

for the parabola is 

2 _s 

p 2 p 2 j9 2 

y s 



Now (XV. 1st) ^2pa? + p 2 = n, which is the normal of the 
point (x, y) of the parabola. Therefore for the parabola r — 



n 3 



-^ ; i. e., The radius of curvature of any point (x, y) of the para- 

P 

bola is equal to the cube of the corresponding normal divided by 

the square of the semiparameter. Now, p being constant, r 

varies directly as n 3 ; and n, which has the minimum value 

= _p, when x = 0, increases with x indefinitely. Hence in the 

parabola, the greatest curvature is at the vertex of the curve 

for which the radius of the osculatory circle is equal to the 

semiparameter; and the curvature of the branches diminishes 

continually as the branches recede from the vertex. 

XXVI. Involutes and involutes. 

Conceive a thread, flexible and inextensible, applied over 
the curve CDB (Fig. 32) so as perfectly to coincide with it 
from C to B. If this thread be gradually removed from the 
curve, so that the removed portion be rectilinear, on the same 
plane as the curve and tangent to the curve, as DM, for in- 
stance ; the extreme point M of the thread thus evolved traces 



110 PRINCIPLES OF INFINITESIMAL CALCULUS. 

out another curve, which is called the involute, as the one from 
which the thread is evolved is called the evolute. From the 
same evolute different involutes may be obtained^ taking 
threads of different lengths, as, for instance, CDL, which, when 
applied to the curve, goes beyond B to A, BA being tangent 
to the curve in B, and the involute corresponding to this 
thread being ALL'A'. Now whatever be the involute ob- 
tained by the evolution of the thread, the centres of the oscu- 
latory circles of the involute are all along the evolute, each 
and all of whose points are centres of these circles. Let, in 
fact, the extremity M or L describe, by the evolution of the 
thread, the infinitesimal arc MM' or LL'. The length of the 
evolved thread, in passing from the first to the second posi- 
tion, varies only by an infinitesimal quantity ; hence the arc 
LL' will coincide with that of a circle described by a radius 
having its centre in D, between the contacts of the first and 
second tangent, and LD or MD for length. Now what we 
say of the infinitesimal element LL' or MM' is applicable to 
* each and all the other elements of the involutes. Hence all 
the centres of the radii of curvature of the same curves are 
along BDC, of which each point is one of them. We come to 
the same conclusion by a different process. Let AA' be any 
portion of curve, whose curvature diminishes from A to A'. 
Supposing that AA' turns its concavity toward the axis AX, the 
tangents of its different points will form angles with AX, con- 
stantly diminishing from A to A'. Now the radius of the 
oscillatory circle of each point of the curve is perpendicular to 
the tangent corresponding to that point ; hence the radii of the 
osculatory circles, corresponding to the points L and L', will 
form an angle with each other, and representing by LM, L'M' 
these two perpendiculars, they will meet somewhere at a greater 
distance from L than D, the centre of the osculatory circle 
corresponding to L, on account of the diminishing curvature 
of AA' toward A'. Now if we take LL' infinitesimal, the 
prolongation of LD, from D to the point of intersection with 



INTEGRAL CALCULUS. Ill 

the normal from L/, is also infinitesimal; hence the same 
point is the centre of the osculatory circle of L/. Observe, in 
fact, that a circle may be described which passes through L 
and L ; , whatever the arc LL/ may be, having its centre some- 
where on the prolongation of LD. But when LL' becomes 
infinitesimal the prolongation of LD also becomes infinites- 
imal, the arc of the circle coincides with the element LL r of 
the curve, and the point of intersection of the two normals 
from L and L' is the centre of this circle, and particularly the 
centre of the osculatory circle corresponding to L'. Following 
the same process for succeeding points, we obtain a polygon 
of infinitesimal sides, the points of concurrence of which are 
centres of osculatory circles of the different points of the curve 
A A'. But a polygon of infinitesimal sides coincides with a 
curve, and each infinitesimal side coincides with a tangent to 
this curve. Again, this same side, produced, forms the radius 
of the osculatory circle of the point of the given curve met by 
it. Hence the centres of the osculatory circles of AA', which 
represents any curve different from the circle, form another 
curve BDC, and the radii are the tangents of this curve taken 
from the points of contact to the points of AA' met by them, 
the points of contact being the centres. 

The law with which the curvature varies is different for dif- 
ferent curves ; hence each curve has its own evolute. Let us 
see two examples in the evolute of the parabola and in that of 
the cycloid. 

I. We have seen in the preceding paragraph (XXV.) that 
the radius of the osculatory circle of the vertex of the parabola 
equals the semiparameter p. Xow the axis AX (Fig. 33) of 
the parabola is perpendicular to the tangent corresponding to 
the vertex A. Taking, therefore, on the axis, AB = p, B is 
the centre of the osculatory circle of the vertex A, and one of 
the points of the evolute ; the axis AX is besides a tangent of 
the evolute in B. Taking now any point L of the upper 
branch, whose co-ordinates are x = AH, y — HL, let LN be 
8 



112 PRINCIPLES OF INFINITESIMAL CALCULUS. 

the normal corresponding to that point, and (XXV.) taking 

LD = yK 2 — — — , D is the centre of the osculatory circle 

corresponding to L. 

To find the equation of the evolute let us refer it to the 
orthogonal axes having their origin in B, and the axis of 
abscissas coinciding with the axis AX of the parabola. Rep- 
resenting by x, y y, the co-ordinates of the evolute, drawing from 
D, DH' perpendicular to AX, we shall have for the point D, 
x, = BH', y, = H'D. Draw from D, DD, parallel to AX, 
and produce LH until it meets this parallel in D,. Let, lastly, 
N be the point of the axis AX, met by the radius LD. LIST 

is the normal of the point L which (XV. 1st) is equal to 

i 
{2px + p 2 ) 2 , HN is the subnormal and (ibi) = p. Xow from 
the similar triangles LHN, LDD, we have LX : LD : : XH : 
DD„ LX : LD : : LH : LD, ; i. e., 

3_ 

{2px + p 2 ) Y : (^P1±PV_ . :p . HH'; 
P 

(2px + pY : ( 2 P X + P 2 ) 2 : : (2p X y . LD, ; 
p 

from which 

HH' = 2x + p, LD, = \/fyx~ -f — ^2^x~; 

and consequently, since BH' (= x f ) — HH' — HB = HH' -f 
AH — AB, and DH' (= y,) = LD, — LH; 

2x 



x, = 3x, y, — — *^2px. 

Substituting in the second of these equations the value of x 
taken from the first, and squaring the members of the result- 
ing equation, we obtain 

2 _ 8 3 



INTEGRAL CALCULUS. 113 

the equation of the evolute of the parabola of the second order, 
which is itself called a parabola, but of the third order or 
cubic. It has two symmetrical branches, one on each side of 
the axis AX from B toward X, the branches turning their 
convexity to the axis, and ending in a cusp at B, where AX 
is a tangent to both branches. 

ii. We have (XXIV. I.) for the cycloid, ^ = \J^Zl f 

taking the origin of the orthogonal axes in the vertex, or ex- 
tremity, A of the axis AA' (Fig. 34). But let us take the 
origin of the orthogonal axes at the extremity B of the base 
BB', taken as axis of abscissas, BY', perpendicular to the base, 
being the axis of ordinates. Let now M be any point of the 
cycloid, whose co-ordinates x', y', with reference to the new 
system, are BK', K'M. Now BA' = *c, c being the radius 
of the generating circle ; hence MK, or the ordinate y of M, 
referred to the axes AX, AY, is equal to A'B — x' = &k — x'i 
the abscissa of M referred to the same axes or x = AA' — A'K 
= 2c — y'. Hence dx = — dy', dy = — dx f . Therefore, sub- 
stituting these values in the above equation, we obtain 

dx' V —J, — 

n + f n (x)~\ 2 

Now the first of the two formulas y y 1 +/ /2 (#), t — J \ \ t 

f \ x ) 
gives (XV.) the value of the normal n, and the second gives 

(XXV.) the value of the radius r of the osculatory circle of 
any point of a plane curve. In our case/' 2 {x r ) — (~f~ f Y = 

~- = — 1 -j 7, and this equation differentiated gives 

y y 

2/' (x')f" (x) dx' = —%% dx' = —^f (x') dx' ; hence 
J v JJ v ' y' 2 dx' y 

f» ( x ') = — 4 2 . Therefore (XV) 



114 PRINCIPLES OF INFINITESIMAL CALCULUS. 

n = MN = y f <Sl -f f 2 (x f ) = \/2by' 

hence r == 2n ; i. e., tlie radius of curvature of any point of the 
cycloid is the double of the normal of the same point. Now 
the normal corresponding to the origin B is = 0, and the 
normal corresponding to the vertex A is = 2c ; therefore the 
radius of curvature corresponding to the origin is = 0, and the 
radius of curvature corresponding to the vertex is = 4c. Pro- 
ducing therefore AA' to A, so as to have A' A, = 2c, and pro- 
ducing the normal MN of the point M to D, so as to have 
ND = MN, the evolute of the semicycloid BMA is a curve 
which passes through the points B, D, A,. To determine the 
quality of this curve, let the arc BD = tf, and referring the 
evolute to BA r taken for axis of ordinates, and to BX ; , taken 
for axis of abscissas, and representing by x l} y l the co-ordinates, 
we shall have, with regard to the point D, x x = BK, — DD„ 
and y l = DK,. Now, since ND = MN, the triangles NMK', 
NDD, are equal ; hence MK ; = DD,, i. e., y' — x, ; but the 
arc BD = MD, and MD == 2^2^/, therefore 

tf = 2^2^ 

Now this value belongs (XXI Y. I.) to a cycloidal arc having 
for axis BB r = 2c = AA'. Therefore the evolute BDA, of 
the semicycloid BMA, is another evolute equal to the latter, 
but inverted. In like manner, the other semicycloid B'A has 
the corresponding evolute B'A', which is again another semi- 
cycloid equal to B'A. 

XXVII. Integration by series. 
When a differential /' (x) dx cannot be accurately integrated, 
the integration may be obtained by means of a series as near 
as desirable to its exact value, provided the conditions, which 
we here subjoin, be verified. 



INTEGRAL CALCULUS. 115 

Let Xi, X 2 , X 3 . . . represent different functions of the vari- 
able x, and let f (x), from x = x to x = x m , be capable of 
being developed into a converging series X x + X 2 + X 3 -f . . . , 
i. e., into a series the terms of which diminish in such a manner 
that, by increasing indefinitely their number, the sum of all 
approaches ever more to the fixed and determined limit f r (x). 
In this supposition we shall have 

/??/' (x) dx =*/*■ X, dx + f Xm X 2 dx + /*» X 3 dx + . . . 

To simplify the case, let X 2 = A, X 2 = B.r, X 3 = Cx 2 , etc. 
The preceding formula will be changed (XXIII. (l), (n.),) 
into the following : 

A Xm f (*) dx = A (x m — x ) + jB (xj — x 2 ) + }C (xj — x*) 

Xq 

. + . . . 

Consequently, making a? = 0, and taking x for x mf 
f*f (x) dx = Ax + iBx 2 + JCb 3 + . . . 

And this series represents the definite integral of f f (x) dx 
between x = 0, and x, approaching more and more to its exact 
value the greater is the number of terms that are taken. 

By this method of integration we may develop into series 
those functions which are expressed by definite integrals. 
Let us see it exemplified in the following cases ; and 

1st. Let log (1 -f x) be a given function of x. From the 
first case (XXIII.) we have log (1 + x) — log (1 + x ) 

x dx 
= f ' — ; and consequently, with x = 0, 

Xq I ~\~ X 

log (1, + *) = f *L = f _L_ dx. 

\j I -\- x \) 1 + % 

Now (see Alg. § 67), supposing x < 1, = 1 — x + x 2 — 

1 + x 

x % + . . . ; the series of the second member being unlimited, the 

condition therefore to be verified, in order to have f\ - dx 

01 + a; 



116 PRINCIPLES OF INFINITESIMAL CALCULUS. 

expressed by a convergent series, is verified, provided x < 1 ; 
i. e., in this supposition, 

_•.£/ (JiJu X X X . 

therefore within the same limits, 

/y2 /V,3 ,y,4 

*(! + *).-»* f + |-|+" • 

2d. Let the given function be arc (sin = a?). Now (VI. Vii.) 

d arc (sin = x) = — — ■, therefore 
i/l-^' 

/I — = arc (sin = x). 

But (XL 2d,)if»<l, 

"111 ~2 i 4 i O O g . 

== "• 2^*^' "7™ 77 7 3? ~| 77 Z 7» *^ "T" • • • 



s/\~Ttf * '2-4 ' 2-4-6 

Hence 

arc (sin = a?) ==/^ (l + |* 2 + — ^ + — — x 6 + . . .) dx 

|1 3 i " 5 i o'o 7| 

By means of this series we may find the value of the semiperi- 
phery nc of the circle having 1 for radius as nearly as desirable. 

if 

For. take x = J, the corresponding arc is -- ; hence 

r = Q(l + L_ j_ ?_ l 3 ' 5 4. 

v 2 ^2-3-2 3+ 2-4-5-2 3 ^2-4-6-7-2 7 ^ "* 

= 3, 1415926 . . . 
3d. Let also the given function be arc (tg = x). We have 

(VI. ix.) d • arc (tg = x) = ^ > hence 

J. - j~ X 

But if x < 1, — L-, = 1 — a; 2 + x 4 — a 6 + # 8 — . . . 
1 -f a; 2 



INTEGRAL CALCULUS. 117 



ig 

Now the arc of a tangent < 1 is necessarily < — . There- 
fore, for the positive arcs from 0° to — , we have 

x x 

arc (tg — x) =f~ (1 — x 2 + # 4 — a; 6 -|- a? 8 — . . .) dx = x — *— 
o 

X 5 X 7 

+ 5~~ 7 + *•• 
The complement of arc (tg = x), when x <C 1, is an arc 

between --- and —. Representing by z its tangent, the value 
of this tangent ranges from z == 1 to z = o> ; hence — <] 1. 
Also arc (tg = 2) = — — arc (tg = #), and since tg a == 

are (tg = 2) = -| — arc (tg = -) ; 

and arc (tg = --) = _- _ + __- + ... 

Therefore for the positive arcs from — to — we have 

4 2 

u * 1 1 1 1 

arc(tg = , ; = --- + -3-^ + ^-... 

XXVIII. Integration of differential equations of the first order 
and. degree, and between two variables. 

An equation between two variables is an equation in which 
enter only two variables and their differentials. To integrate 
differential equations means to find a finite equation between 
the variables of which the differential is a result. The order 
of the equation is the same as that of the differential of the 
variable taken as function of the other. For instance, the 



118 PRINCIPLES OF INFINITESIMAL CALCULUS. 

variables being y and x, if y be considered as function of x, if 
the differential of y entering in the equation, be of the second 
or third order, the equation also would be said to be of the 
same order ; and since the order of the derivative follows that 
of the differential, so also the order of the equation is taken 
likewise from that of the derivative of the same function. The 
degree of the differential equation is taken from the power of 
the differentials, either dx or dy, or both. We limit our dis- 
cussion on differential equations between two variables, to 
those only of the first order and degree, such as 

0) <p (*, y)dx + x fo y) dy = p. 

Now the equation (g) may be the result of the differentiation, 
for instance, of / (x, y), which we shall represent by fx, or the 
elements contained in {g) may be combined together otherwise. 
In the first of these cases (g) is said to be an exact or total dif- 
ferential ; in the second, it is simply called a differential equa- 
tion, or, to distinguish better this case from the other, we 
may call it inexact differential. We shall consider the two 
cases separately, and, commencing w T ith the exact differential 

first, since (XVI.) dp = d x p -f d y p or = -=- dx -f y dy, and 

the differential of p is by supposition the first member of (g), 
Ave shall have 

t v dp f \ dp 

but from these equations we obtain 

d 9 (%, y) _ d 2 p dx{x,y) _ d 2 p 
dy ~ dxdy' dx dydx 

and (XVII.) -=-^- = -r^r- ; hence, also, 
dxdy dydx 

. d 9 (x, y) _ d x (a, y) 
Ky,) dy " dx 
Therefore, in order that (g) may represent an exact differential, 
the equation (g f ) must be verified. 



INTEGRAL CALCULUS. 119 

Now, to find out the original function /(a?, y): Since the first 
term of (g) is the partial differential off (x, y), relatively to x ; 
by integrating this term as a differential function of x only, we 
shall obtain / (x, y). But an indefinite integral may be ex- 
pressed (XXIII. i.) by an integral, which begins by a particular 
value of the variable, with an arbitrary constant added to it. 
Designating then by Y an arbitrary function of y, which in 
the present case is regarded as constant, we shall have 

«• =f~ <P ( x > y) dx + Y - 

To determine Y, let us differentiate this equation relatively 
to y. We shall have 

***** = fx% 9 & y + dy ^ dx ~~/x ^ $ dx + dY > 

and (XXI. 2d) 

, r xd(p(x,y) , , ! jxr 

dut*> —f ; lr* dydx + dY. 

x dy 

But d^ = x (x, y) dy, and, since from {g) ^ ' y ' = -2£Lj-d}- } 

LtU iXJU 

therefore 

X fo V) dy = f c ±*L x jll dxdy + dY 
Xq clx 

f dx ^/ y) dxdy + dY 
Xn clx 

= x ( x , y)dy — x (ft, y) dy + d Y ; 

and consequently, 

d Y = x Oo, y) dy, Y = f y x ( x o, y) dy + c. 

ffo 

Substituting now this last value of Y in the first of the above 
equations, we shall obtain 

(#2) f* = /f <P {x, y) dx + f y x ( x o, y) dy + c, 
*°o yo 

in which c represents, as usual, an arbitrary constant. 



120 PRINCIPLES OF INFINITESIMAL CALCULUS. 

Let, for example, 

(Qxy — y 2 ) dx -f (3a; 2 — 2xy) dy = 0, 

in which <p (x, y) = Qxy — y 2 , x { x > y) — 3# 2 — 2xy, and con- 
sequently, 

lifej/) _ 6a . __ 2v = d *( x >y) m 

dy U dx 

The condition (#,) is thus verified, and the given equation is 
an exact differential to which the resolution expressed by (g 2 ) 

may be applied. Now f (Qxy — y 2 ) dx — 3x 2 y — 3aA — y 2 x 

Xq 

+ f x ^fl ( ; W — 2x y) dy = 3avfy — Sx 2 y — a^ + a? 2/ a 2 ; 

hence 

p. = Sa?y — tfx — (Sxfy — x y 2 ) + c. 
The terms 3xfy Q — x Q y 2 are constant, and, observing that the 
given equation has for the second member, its integral 
\k also must be a constant. Representing now by C the sum 
of all these constants, we shall have 

Sx 2 y — y 2 x = C, 

which, differentiated, reproduces the given equation. 

Inexact differential ; resolution by multiplication. Let us now 
take (g) as representing any differential equation of the first 
order and degree between two variables. Its integral may be 
obtained by different methods, one of which is to find a fac- 
tor M, by which multiplying (^), the product will be an exact 
differential. This factor, however, is not easily found, except 
in the two instances which alone we shall examine here. 

First case. The first of these two cases occurs when the 
factor M, which renders (g) an exact differential, is a function 
of x only or of y only. But if (g) multiplied by M becomes 
an exact differential, the condition expressed by (g f ) must be 
verified about this product ; i. e., we must have 

rf[M<p(tt,y)] = rf[M x foy)] _ 
dy dx 



INTEGRAL CALCULUS. 121 

Now (VII. ii.) 

d[M,(x y y)-] = M d 1 {x ll ) ( dM 
dy dy yv ^ ; % , 

drM.x(x,y)~] ^dxfay) , ( \dM. 

Therefore 

, r d (p (a;, «) , , c?M , .. c? y (x, y) , , dM. 

M dy + ? (*' ^ % = M — ^ + * ( *' y) & • 

But in the supposition of M being function of the only varia- 

/7TVF 

ble x, -=— = 0, as in the supposition of M being function of 

dM. 

the only variable y, -j— = 0. Hence in the first of these sup- 

positions the condition of integrability of (g) x M is, that 
1_ dM = _ 1 , d(p(x,y) __ dx{x,y)\. 
M dx x{ x )V) dy dx ' 

in the second, 

_L ^ = 1 ( d x( x >y) _ d v{%>y) \ 

M dy 9 (x, y) ^ dx dy ' 

Now -=-r . — — is a function of the only variable x. and ^-=- . — — 
M dx J 9 M dy 

is a function of the only variable y ; hence M being a function 

of the only variable x, (g) X M cannot be an exact differential, 

unless 

1 / d<b(x,y) __ dxix,y)\ 

X {x, y) \ dy dx ' 

be a function of the only variable x. And M being a func- 
tion of the only variable y, (g) X M cannot be an exact differ- 
ential, unless 

1 /dxfay) dcp(x ,y)j 
cp (x } y) ^ dx dy ' 

be a function of the only variable y. Supposing now that the 
one or the other of these conditions is verified, the factor M 
remains to be found. 



122 PRINCIPLES OF INFINITESIMAL CALCULUS. 

Calling, for brevity's sake, x> <P the functions x { x , y)> <P (#, y), 
from the last two equations, we have 

dM 1 fd<p dx\ , dM 1 ,dx cftp\ , 
"M ~ X ^dy"d^ ' M" = 7 (ch~~~dy> J ' 

Now (VI. i.) ^=dlog(M). Therefore 



^w=/^(g-|-)^ 



logXM)=/i(g-*)^ 



1 /dx df 

<p ^dx dy 



i. e., in the first case, 

M = e * y 
in the second, 



1 /Jx <f/i 
Let, for example, 






— <ia? -. c?v = 

2/ 2/ 2 

2 x 
be the given equation, i. e., let <p = — , x — 2 » with these 

elements, 

X dy dx* x ^ 2/ 2 y 2 ' a? ' 

that is, — (-= =?) is a function of the only variable x 

X y dy dx' 

1 /d(f> d\\ f dx 

And the factor M = e = e : but — = 

c£ log a?, therefore M = e los (*) = a?. With this factor the given 
equation becomes 

— dx 5 cfy = 0, 

2/ 2/ 

whose first member being an exact differential, its integral is 



INTEGRAL CALCULUS. 123 

obtained by means of the formula (g 2 ), from which, making 

x = 0, we obtain 

r 2 

- = C, 

y 

which, differentiated, reproduces the preceding equation, from 
which we obtain the given one, dividing it by x. 

Second case. The second case in which we propose to find 
the factor M, is when the equation (g) is homogeneous, and 
both functions 9 and •% of the same degree. 

We call homogeneous a function f (x, y) y in which the terms 
are reducible to an integral form and are all of the same dimen- 
sion, as in the following trinomial : 

6x 5 + y 3 x 2 — 2y 4 x, 

in which the sum of the exponents of the two variables is of 
the same dimension or degree, 5 in each term. 

Supposing n to be the degree, and multiplying each variable 
by any factor u, we shall evidently have / (ur, uy) = u n f{x, y). 
Considering now u as a variable, and, differentiating the last 
equation with regard to u alone, we shall obtain 

nu n ~ 1 f(x,y) du = f' ux (ux } uy) dux +f uy {ux, uy) duy } 
— xf'ux (ux, uy) du + yf'uy (ux, uy) du ; 
from which, making u = 1, 

nf fo y) = »/'. {%, y) + yfy fe y) ; 

i. e., The product of the homogeneous function f (x, y), by its 
degree n, is equal to the sum of the products of the partial deriva- 
tives, by the variable to ivhich the derivatives are referred. 

Let now M be another homogeneous function of x and y, 
and such, that multiplying (g) by it, the product be an exact 
differential of ^, and /x itself be a homogeneous function of 
the nth degree of the same variables. We shall have 

M 9 dx + M x dy = d[x • 
and from the above theorem, 

Ma; 9 + My % = n V" 



124 PRINCIPLES OF INFINITESIMAL CALCULUS. 

Calling h the degree of M, and h that of <p and x? the degree 
of fx must necessarily be Jc + h + 1 ; i. e., n == 7c -{- h -\- 1. 

Divide now the first by the second of the last two equations, 
we shall obtain 

odx -f xty 1 d\u 

XCp _J_ y x n ' p ' 

-j 7 -1 

Now (VI. 1. ) — . — = — dl (fx) : hence the second member 

of this equation is an exact differential, and consequently the 
first also ; but the first member is the product of (g) by M = 

; hence whenever 9 and y are homogeneous functions 

x$ + yx 

of the same degree as x and y, the equation cpdx + x<% — be- 
comes integrable by being multiplied by M = . 

Let, for example, 

(xy + y 2 ) dx — x 2 dy = 
be one of the equations represented by (g) in our present sup- 
position. The factor M will be —. ^ _ = — •„, by 

x x [xy -f y) — yx xy 7 

which multiplying the given equation, we obtain 

( 1 ) dx „ d,y = ; 

the first member of which being an exact differential, it may be 
integrated by means of the formula (g 2 ), from which, taking 
x — 0, we obtain 

for the integral corresponding to the given equation. 

Resolution by separation. Besides the method of multipli- 
cation, the integral of an incomplete differential may be ob- 
tained by separating, when possible, the variables of the func- 
tions, so that, representing by X and Y two functions, the 
first of the only variable x, the second of the only variable y, 



INTEGRAL CALCULUS. 125 

the equation (g) may be reduced to the form ~Kdx + "Ydy = 0. 
An equation of this form can be integrated with the rules 
ordinarily applicable to differential expressions. Now this 
separation can be obtained easily in the two cases which we 
propose to examine here. 

First case. The first of these cases occurs when (g) is homo- 
geneous ; for, making y = zx, and substituting this value in 
(g), which we shall suppose of the nth. degree, <p and x w ^ 
each have x n for common factor, and (g) will thus become a 
function of the only variable z ; for, from <pdx + xty = 0, we 

infer — dx + dy = 0, in which the ratio — (we shall call it Z) 

X X 

is a function of the only variable z. Now, from y = zx we 

have dy = zdx + xdz ; hence 

Zdx + zdx 4- xdz = ; 
and therefore 

dx dz _ 

r- Tj—, — = u> 

x L + z 

which is an equation with separate variables, and whose in- 
tegral is 

in which substituting — for 2, we shall obtain the finite equa- 
tion between x and y corresponding to the given(^). 

Let us see an example in the following homogeneous equa- 
tion of the second degree : 

(xy — y 2 ) dx — (xy + x 2 ) dy = 0. 
Making in it y = zx, and substituting zdx + xdz for dy, it 
will become 

{z — z 2 ) dx — (z -f- 1) (zdx -j- xdz) = 0, 
or 2z 2 dx + x (z + 1) dz = 0, 

easily reduced to the following : 



126 PEINCIPLES OF INFINITESIMAL CALCULUS. 

dx Z -f 1 7 

with separate variables; which, integrated, gives us 

lo g (*) + ilog (s) — — = C, 

and consequently the finite equation 

log^).+ ilpg(|)-i = C, 

corresponding to the given differential. 

Second case. The other case, in which the variables of (g) 
can easily be separated, is when 9 and x are Slicn functions of 

x and y, that the ratio — results equal to a product X * Y, in 

X 

which X is a function of the only variable x, and Y a function 

of the only variable y ; for (</), or its equivalent — dx -f dy — 0, 

then becomes X ■ Ydx -f c?// = ; and from this we obtain 

Xdx -f $ = 0, 
Y 

with separate variables. 

Let, for example, the differential equation be 

for which the condition 

X #?/ -f &* 3 y 1 -f .t 2 

dy 
is verified. Consequently we have ~Xdx -f -^ = 0, or 

dy dx _ 

/- -1 1 2 J 
v/^ 1 + or 

whose indefinite integral is 

2*Sy — arc (tg = x) — C. 

In other cases, when the separation of the variables is pos- 



INTEGRAL CALCULUS. 127 

sible, the resolution is obtained bv means of substitutions for 
which no general rule can be assigned. Some analytical pro- 
cesses also can be employed, with advantage, to the same effect. 
An example of this kind may be seen in the following 
number. 

XXIX. Integration of linear differential equations of the first 
order , containing only tivo variables. 

We call linear equations those in which the dependent 
variable y, and its differential dy, do not exceed the first 
degree, and are not. multiplied by each other, whatever the 
degree of the other variable x may be. Hence, representing 
by X, X 1? X 2 different functions of the only variable x, the 
equation XcZy -f- !L x ydx + X 2 cfo = 0, or its equivalent 

X | + Xl2 , + X 2 =0, 

is the general formula of all linear equations of the first order, 
between the variables x and y. Representing by / (x) and 

X X 

cp (x) the ratios — , ■—-, the same formula may be expressed 
X X 

also by 

(L) dy -f yf (x) dx + <p (x) dx = 0. 

The integration of (L) is obtained by the separation of the 
variables ; and this separation by means of the following ana- 
lytical process. 

Let u, z be two indeterminate functions of x ; such, however, 
that we may have y = w z, and consequently dy = udz -f zdu, 
which value substituted in L gives us 

udz + z {du + uf (x) dx) + cp (x) dx = 0. 
Xow, u is an arbitrary function which can consequently be 
determined in such a manner as to render the binomial du -{- 
uf (x) dx = 0. Thus the last equation may be resolved into 
the two following : 

du 4- uf(x) dx = 0, udz -f- 9 (x) dx = 0. 



128 PRINCIPLES OF INFINITESIMAL CALCULUS. 

The first of these equations divided by u, and integrated, 

gives 

log (u) = — ff(x) dx = log (g -//(*) dx ), 

in which we omit the constant, being included in the indefinite 
integral of/ (x) dx. Now the last formula is equivalent to the 
following : 

in which u is given by a function of x, and such a function as 
to verify the condition du -f uf(x) dx = 0. But the general 
formula, as we have seen above, becomes, in this case, udz + 
cp (x) dx ==■ 0, from which, substituting in it the value of u just 
found, we obtain 

dz = — 9 (x) e-ffW dx dx, 
with separate variables z and %. This formula, integrated, 
gives 

z=y~= —fcp (x) effW d * dx -f C, 
u 

or (L,) y = e ~ff (*> dx [C — / <p (x) e// ( x ) da; cfor], 

which is the integral of the general equation (L). 

Let, for example, the following differential equation be given : 

dy + xydx — xdx = ; 
or, comparing it with (L), \etf(x) = x, 9 (x) = — a?, and con- 

sequently /jf (#) dx — fxdx = *— + c, and from (L J 

y = e~ * (Ce- C + fxe* dx) = <f *(C, +fde~-\ 
or y = 1 + — I -. 

C 2 

XXX. Integration of linear differential equations of the 
second order, and between two variables. 

"We shall limit our resolution to the equations represented by 



INTEGRAL CALCULUS. 129 

in which a, a' are constant coefficients. To obtain the resolu- 
tion, let us take the auxiliary equation 

R 2 + aR + a! = 0, 
and let r, r f be its roots. We shall have (Treat, on Alg. § 99) 
a — — (r + i' f ), a? = rr', and consequently the given equation 
may be changed into the following : 

d 11 du 

dx 2 "~ ^ + T,) ctx + Vffy = X ^' 

d 2 y dy . ,dy N . N 

(% l _£. ipy } 

AT d 2 y dy ^dx ' , , . dy , 

JN ow ~ — r -f^ = i ? hence, making -f — ry = y, 

dx 2 dx dx ' to dx J J 

(E) may be furthermore changed into 

which is a linear equation of the first order. Thus the inte- 
gration of (E) is obtained by means of the integration of the 
two 

( dy' . . / x 

(E<) i X 

both of the first order, and both easily reducible to the form 
(L) of the preceding number, as follows : 

dy f = y'r'dx + x ( x ) dx y 
dy = yrdx + y'dx, 

which, compared with (L), and resolved by means of (L,), 
give 

y f =-. e r ' x (C + fx xe ~ rx d&)i 



^ 2 ' [y= e rx (C + fy f e- rx dx). 
The relation between x and y, resulting from these two 



130 PRINCIPLES OF INFINITESIMAL CALCULUS. 

equations, belongs to the given differential equation (E), and 
is consequently its complete integral. 
To see an application, let 

be a linear equation of the second order to be integrated, and 
let us take it, first, with the upper sign. Its auxiliary equa- 
tion will, in this case, be R 2 — b 2 = ; and, consequently, r = 
b, v' — — b ; also x ( x ) = 0. Thus the formulas (E 2 ) will 
become 

y r ^ e -bx • C, y = e hx (C -f fy'er bx dx), 

= e bx (C + Qfer 2bx dx), 

= e h *(C' — ~fe- 2bx log e d (— 26a?)) . 

Now (VI. I.) e~ 2bx log e d (— 2bx) = de- 2bx ; hence 

y = e hx (C' — £ e~ 2to ) = CV* — C"e- & * 

is the integral of the given equation relatively to the upper 
sign. Let us now take the positive sign : We shall have 
R2 + h 2 = 0> Hence r = 5s/_i ? j./ = _ 6n/ — 1, x (a?) = 
; i. e., everything as with the negative sign, except the 

change of b into b^ — 1 ; hence, regarding the imaginary 
quantities as real ones, we shall have for the integral of the 
given equation, taken with the lower sign,' 

C 

2b s/~ 



y = c >-*f-i(C'-_^ L=e-^-i) ; 



26v/ZTi 6 

c 

and, making 6 = 1 and- ^, = . = C 2 , 

2/ = C' c xV ~ — Q 2 e~ xV ~\ 
Now from Maclaurin's theorem we have 



INTEGRAL CALCULUS. 131 



X 2 X 3 



e*=l + *+_ + _ + ..., 



sin x — x — — — + 



x 3 x 5 



2-3 ' 2-3-4-5 



x 2 x* 



cos X = 1 — 1 



2 ' 2-3-4 

Taking ± x^ — 1 in the first of these series instead of x y 
from this substitution, and from the other two series we obtain 

e x 1 = cos x + sin x*S — 1, 

e~ x x = cos x — sin x V — 1 • 
hence 

y = C (cos a; -f sin a;*' — 1) — C 2 (cos x — sin x >/ — 1), 
= (C — C 2 ) cos x + (C — C 2 ) v/37i • s i n a-. 
Calling K the difference C — C 2 , and K, the product 
(C — C 2 )Y — 1; we shall finally obtain 

y = K cos # -f K, sin a?. 



THE END. 





X 



TCv X 



Fig 



X X- 



?ig. 16. 



T \ 



D 



u 



ri.i. 




A Kl" K. K 1 



Litk, by D BITTER &ALTO. 



Fig. 20. 



J¥ 



Y 



T? A 



K K X 



- 






PL.IL 




Lith. by D. BITTKR.. B ALTO. 



